Answer:
a ₙ = 7n
Step-by-step explanation:
This is an arithmetic sequence, the common difference between each term is 14-7 = 21-14 = 28-21 = 7
to the previous term in the sequence addition of 7 gives the next term.
Arithmetic Sequence:
d = 7
This is the formula of an arithmetic sequence.
a ₙ = a₁ + d(n − 1)
Substitute in the values of
a₁ = 7 and d = 7
a ₙ = 7 + 7(n − 1)
a ₙ = 7 + 7n -7
a ₙ = 7 - 7 +7n = 7n
a ₙ = 7n
Answer:
Step-by-step explanation:
In an arithmetic sequence, consecutive terms differ by a common difference and it is always constant. Looking at the set of numbers,
14 - 7 = 21 - 14 = 28 - 21 = 7
Therefore, it is an arithmetic sequence with a common difference of 7.
The formula for determining the nth term of an arithmetic sequence is expressed as
Tn = a + (n - 1)d
Where
a represents the first term of the sequence.
d represents the common difference.
n represents the number of terms in the sequence.
From the information given,
a = 7
d = 7
The function that represents the sequence would be
Tn = 7 + (n - 1)7
Tn = 7 + 7n - 7
Tn = 7n
The relationship between the number of games won by a minor league baseball team and the average attendance at their home games is analyzed. A regression to predict the average attendance from the number of games won has an r = 0.73. Interpret this statistic.
Answer: There is a strong positive correlation between number of games won by a minor league baseball team and the average attendance at their home games is analyzed.
Step-by-step explanation:
The Pearson's coefficient 'r' gives the correlation between the predicted values and the observed values .
It tells the direction and the strength of the relation.When r is negative it means there is a negative relationship between the variables .When r is positive it means there is a positive relationship between the variables .When |r|=1 , strong correlation , When r=0 , there is no correlation.If 0.70<|r|<1 , there is a strong correlation.If 0.50<|r|<0.70 , there is a moderate correlation.If 0.30<|r|<0.50 , there is a low correlation.Given : A regression to predict the average attendance from the number of games won has an r = 0.73.
Since r=0.73 is positive and 0.70 <0.73 <1 , it means there is a strong positive correlation between number of games won by a minor league baseball team and the average attendance at their home games is analyzed.
A roofing contractor purchases a shingle delivery truck with a shingle elevator for $42,000. The vehicle requires an average expenditure of $9.50 per hour for fuel and maintenance, and the operator is paid $11.50 per hour. Write a linear equation giving the total cost C of operating this equipment for t hours. (Include the purchase cost of the equipment.)
Answer:
[tex]T(t) = 42000 + 21t[/tex]
Step-by-step explanation:
There are two costs for the operation of the equipment, the purchase cost and the hourly cost. The total cost is the sum of these two costs:
So
[tex]T = F_{c} + H_{c}[/tex]
In which [tex]F_{c}[/tex] is the fixed cost and [tex]H_{c}[/tex] is the hourly cost.
Fixed cost
A roofing contractor purchases a shingle delivery truck with a shingle elevator for $42,000. This means that the fixed cost is 42000. So [tex]F_{c} = 42000[/tex]
Hourly cost
The vehicle requires an average expenditure of $9.50 per hour for fuel and maintenance, and the operator is paid $11.50 per hour. So for each hour, there are expenses of 9.5 + 11.5 = $21.
So the hour cost is
[tex]H_{c}(t) = 21t[/tex]
In which t is the number of hours
Total cost
[tex]T = F_{c} + H_{c}[/tex]
[tex]T(t) = 42000 + 21t[/tex]
A space vehicle has an independent backup system for one of its communication networks. The probability that either system will function satisfactorily during a flight is 0.985. What is the probability that during a given flight (a) both systems function satisfactorily, (b) at least one system functions satisfactorily, and (c) both systems fail?
Answer:
(a) 0.970225.
(b) 0.999775.
(c) 2.25 x [tex]10^{-4}[/tex].
Step-by-step explanation:
Given the probability that either system will function satisfactorily during a flight is 0.985.
(a) To calculate the probability that both systems function satisfactorily we are given the probability of either system will function of 0.985 which means that both function have probability of functioning satisfactorily of 0.985 each i.e.,
= 0.985 x 0.985 = 0.970225.
(b) The probability of first function will not function satisfactorily is 1 - 0.985 because the probability of first system function satisfactorily is 0.985.
Similarly, probability of second function will not function satisfactorily is also 1 - 0.985 = 0.015
So Probability that during a given flight at least one system functions satisfactorily = 1 - both system will not function satisfactorily
= 1 - (0.015 x 0.015) = 0.999775.
(c) Probability that during a given flight both systems fail or not function satisfactorily = (1 - 0.985) x (1 - 0.985) = 2.25 x [tex]10^{-4}[/tex]
Answer:
(a) P (Both systems functioning satisfactorily) = 0.970
(b) P (At least one system functions satisfactorily) = 0.999
(c) P (Both the systems failing) = 0.00023
Step-by-step explanation:
Let A = the system in use is working satisfactorily and B = the backup system is working satisfactorily.
The probability that either of the systems works satisfactorily is,
P (A) = P(B) = 0.985
(a)
Both the events A and B are independent, i.e. [tex]P(A\cap B)=P(A)\times P(B)[/tex]
Compute the probability that during a given flight both systems function satisfactorily as follows:
[tex]P(A\cap B)=P(A)\times P(B)[/tex]
[tex]=0.985\times0.985\\=0.970225\\\approx0.970[/tex]
Thus, the probability of both systems functioning satisfactorily is 0.970.
(b)
Compute the probability that during a given flight at least one system functions satisfactorily as follows:
P (At least one system functions satisfactorily) = 1 - P (None of the system functions satisfactorily)
= [tex]1-[P(A^{c})\times P(B^{c})][/tex]
[tex]=1-([1-P(A)]\times [1-P(B)])\\=1-([1-0.985]\times [1-0.985])\\= 1-0.000225\\=0.999775\\\approx0.999[/tex]
Thus, the probability that during a given flight at least one system functions satisfactorily is 0.999.
(c)
Compute the probability that during a given flight both the systems fail as follows:
P (Both the systems failing) = [tex]P(A^{c})\times P(B^{c})[/tex]
[tex]=[1-P(A^{c})]\times [1-P(B^{c})]\\=(1-0.985)\times (1-0.985)\\=0.015\times 0.015\\=0.000225\\\approx0.00023[/tex]
Thus, the probability that during a given flight both the systems fail is 0.00023.
"Consider the following argument:
a. George and Mary are not both innocent.
b. If George is not lying, Mary must be innocent.
c. Therefore, if George is innocent, then he is lying.
Let ???? be the proposition "George is innocent", m be the proposition "Mary is innocent", and let ???? be the proposition "George is lying"."
1. Write a propositional formula F involving variables g, m, l such that the above argument is valid if and only if F is valid.
2. Is the above argument valid? If so, prove its validity by proving the validity of F. If not, give an interpretation under which F evaluates to false.
College Mathematics 10+5 pts
"Consider the following argument:
a. George and Mary are not both innocent.
b. If George is not lying, Mary must be innocent.
c. Therefore, if George is innocent, then he is lying.
Let ???? be the proposition "George is innocent", m be the proposition "Mary is innocent", and let ???? be the proposition "George is lying"."
1. Write a propositional formula F involving variables g, m, l such that the above argument is valid if and only if F is valid.
2. Is the above argument valid? If so, prove its validity by proving the validity of F. If not, give an interpretation under which F evaluates to false.
Answer
1. [tex]F = ((g\cap \sim m)\cup(\sim g\cap m)\cap(\sim l\to m))\to(g\to l) [/tex]
2. It is valid. See explanation for proof.
Step-by-step explanation
1. Statement a is an exclusive OR of [tex]g[/tex] and [tex]m[/tex]. This is because only one of them is innocent while the other is not. That is [tex]g[/tex] is true and [tex]m[/tex] is false or [tex]g[/tex] is false and [tex]m[/tex] is true.
Statement b is an implication that [tex]\sim l \to m[/tex].
Statement c is another implication that [tex]g\to l[/tex].
The presence of the word "therefore" in statement c means it is an implication from statement a AND statement b. So we have
[tex]F = ((g\cap \sim m)\cup(\sim g\cap m)\cap(\sim l\to m))\to(g\to l) [/tex]
2.
The validity will be proved using a truth table. [tex]F[/tex] is valid if the last column contains only true values i.e. truth values of T.
From the table (in the attached image), [tex]c[/tex] represents the statement a, [tex]d[/tex] represents statement b and [tex]f[/tex] represents statement c. The last column, which represents [tex]F[/tex], is valid as all its entries are T.
To translate the argument into logic notation, one could write it as ((¬g ∨ m) ∧ (¬l → m)) → ((g → ¬l). Upon considering the case where both George and Mary are innocent, and George is not lying, it can be observed that the argument is invalid.
Explanation:1. The propositional formula F can be written as: ((¬g ∨ m) ∧ (¬l → m)) → ((g → ¬l). This translates the given argument into logical notation.
2. This argument is not valid. An easy way to prove this is by considering a scenario where both George and Mary are innocent, hence g and m are both true, and let l be the proposition 'George is lying', so l is false. In this case, the left-hand side of your main implication results true ((¬g ∨ m) ∧ (¬l → m)) = (false ∨ true) ∧ (true → true) = true ∧ true = true), but the right-hand side of your main implication results false (g → ¬l = true → false = false) hence the full implication results false, which makes the argument invalid.
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Bottled water and medical supplies are to be shipped to victims of a hurricane by plane. Each plane can carry 90,000 pounds and a total volume of 6000 cubic feet. The bottled water weighs 20 pounds per container and measures 1 cubic foot. The medical kits each weigh 10 pounds and measure 2 cubic feet.
(a) How many containers of bottled water and how many medical kits can be sent on each plane?
Answer:4000 bottle containers and 1000
Medical kits
Step-by-step explanation:
The total weights the plane can take per trip is 90000lb. It implies that if we multiply the weight of each bottle container by the number of bottle container and add it to the product of the number medical kit and the weight of each medical kit, we will obtain a total weight of 90000lb.
If x is the number of bottle and y is the number of medical kit
20×x+10×y=90000....i
Also
The total volume the plane can take per trip is 6000ft³ It implies that if we multiply the volume of each bottle container by the number of bottle container and add it to the product of the number medical kit and the volume of each medical kit, we will obtain a total volume of 6000ft³.
Recall, x is the number of bottle and y is the number of medical kit
1×x+2×y=6000....ii
Combining equation i and ii and solving simultaneously,
x=4000 units and y= 1000 units in each plane trip
Answer:
Bottled water = 4000.
Medical kits = 1000.
Step-by-step explanation:
Let x = bottled water
y = medical kits
For the mass (in pounds):
20x + 10y = 90000
For the cubic ft:
x + 2y = 6000
Solving equation i and ii simultaneously,
x = 4000.
y = 1000.
Bottled water = 4000.
Medical kits = 1000.
Western Athletic Club International (WACI) owns and operates a chain of fitness clubs. Currently WACI has 675 members at its Collegetown location, a 13% increase over the previous year. Next year, WACI hopes to grow by the same number of members. 66% of its members are women. What percent increase will next year's growth represent?
Answer:
12%
Step-by-step explanation:
let x represent the actual number of members last year.
current members = 675 and with an increment of 13% over previous year
to find the increment
(675 - x) / x = 0.13
675 - x = 0.13x
collect the like terms
675 = 0.13x + x
675 = 1.13 x
x = approx 597 members were there last year
its hope to increase by the same number next year
percent increase = 675 - 597 / 675 = 78 / 675 = 0.12 = 12%
A perpetuity pays $50 per year and interest rates are 9 percent. How much would its value change if interest rates decreased to 6 percent?
Answer:
We conclude that its value change for 277.8$.
Step-by-step explanation:
We know that a perpetuity pays $50 per year and interest rates are 9 percent. We calculate how much would its value change if interest rates decreased to 6 percent. We know that
9%=0.09
6%=0.06
We get
\frac{50}{0.09}=555.5
\frac{50}{0.06}=833.3
Therefore, we get 833.3-555.5=277.8
We conclude that its value change for 277.8$.
The value change for $277.8.
The calculation is as follows:
[tex]\frac{50}{0.09}=555.5\\\\\frac{50}{0.06}=833.3[/tex]
So,
= $833.3 - $555.5
= $277.8
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The binomial formula has two parts. The first part of the binomial formula calculates the number of combinations of X successes. The second part of the binomial formula calculates the probability associated with the combination of success and failures. If N=6 and X=4, what is the number of combinations of X successes?156486!
Answer:
15 is the number of combination of 4 successes.
Step-by-step explanation:
We are given the following information:
We are given a binomial distribution, then probability of x succes is given by
[tex]P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}[/tex]
where n is the total number of observations, x is the number of success, p is the probability of success.
Now, we are given n = 6 and x = 4
We have to evaluate the number of combination of success and failures.
It is given by:
[tex]\binom{n}{x} = \dfrac{n!}{x!(n-x)!}\\\\\binom{6}{4} = \dfrac{6!}{4!(6-4)!} = \dfrac{6!}{4!2!} = 15[/tex]
Thus, 15 is the number of combination of 4 successes.
The number of combinations of 4 successes out of 6 attempts, calculated using the binomial probability combination formula, is 15.
Explanation:In binomial probability, the number of ways to get a specific number of successes is often calculated using the combination formulas. In this case, the number of combinations of X successes would be determined by the combination formula C(n, x), which stands for the number of combinations of n items taken x at a time. In your case, where n=6 and x=4, the required number of combinations (or ways to achieve 4 successes out of 6 trials) is C(6,4).
The formula for a combination is generally given by: n! / [x!(n - x)!]. Plugging the given numbers into the formula, we get: C(6,4) = 6! / [4!(6 - 4)!]. This simplifies to: 720 / (24*2), which equals to 15. So, there are 15 combinations of 4 successes out of 6 attempts.
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Suppose that prices of a gallon of milk at various stores in one town have a mean of $3.94 with a standard deviation of $0.11. Using Chebyshev's Theorem, state the range in which at least 88.9% of the data will reside. Please do not round your answers.
Answer:
($3.61, $4.27) is the range in which at least 88.9% of the data will reside.
Step-by-step explanation:
Chebyshev's Theorem
This theorem states that at least [tex]1 - \dfrac{1}{k^2}[/tex] percentage of data lies within k standard deviation from the mean.For k = 2[tex]\mu \pm 2\sigma\\\\1 - \dfrac{1}{(2)^2}\% = 75\%[/tex]
Atleast 75% of data lies within two standard deviation of the mean for a non-normal data.
For k = 3[tex]\mu \pm 3\sigma\\\\1 - \dfrac{1}{(3)^2}\% = 88.9\%[/tex]
Thus, range of data for which 88.9% of data will reside is
[tex]\mu \pm 3\sigma\\= (\mu - 3\sigma , \mu + \3\sigma)[/tex]
Now,
Mean = $3.94
Standard Deviation = $0.11
Putting the values, we get,
Range =
[tex]3.94 \pm 30.11\\= (3.94 - 3(0.11) , 3.94 + 3(0.11))\\=(3.61,4.27)[/tex]
($3.61, $4.27) is the range in which at least 88.9% of the data will reside.
Answer:
Using Chebyshev's Theorem, the range in which at least 88.9% of the data reside is [$3.82, $4.06].
Step-by-step explanation:
Given:
The mean of the prices of a gallon of milk is, [tex]\mu=\$3.94[/tex]
The standard deviation of the prices of a gallon of milk is, [tex]\sigma=\$0.11[/tex]
The Chebyshev's Theorem states that for a random variable X with finite mean μ and finite standard deviation σ and for a positive constant k the provided inequality exists,
[tex]P(|X-\mu|\geq k)\leq \frac{\sigma^{2}}{k^{2}}[/tex]
The value of [tex]\frac{\sigma^{2}}{k^{2}} = 0.889[/tex]
Then solve for k as follows:
[tex]\frac{\sigma^{2}}{k^{2}} = 0.889\\\frac{(0.11)^{2}}{k^{2}}=0.889\\k^{2}=\frac{(0.11)^{2}}{0.889}\\k=\sqrt{0.0136108} \\\approx0.1167[/tex]
The range in which at least 88.9% of the data will reside is:
[tex]P(|X-\mu|\geq k)\leq \frac{\sigma^{2}}{k^{2}}\\P(\mu-k\leq X\leq \mu+k)\leq 0.889\\P(3.94-0.1167\leq \leq X\leq 3.94+0.1167)\leq 0.889\\P(3.8233\leq X\leq 4.0567)\leq 0.889\\\approxP(3.82\leq X\leq 4.06)\leq 0.889[/tex]
Thus, the probability of prices of a gallon of milk between $3.82 and $4.06 is 0.889.
Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: Shock resistance High LowScratch High 70 9Resistance Low 16 5Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance. Are events A and B independent?
Answer:
Part a
The probability that the event disk has high shock resistance is 0.86
Part b
The probability that a disk has high scratch resistance given that the disk has high shock resistance is 0.8140
Step-by-step explanation:
(a).
From the given information,
Let A denote the event that a disk has high shock resistance,
and let B denote the event that a disk has high scratch resistance.
SHOCK HIGH(A) SHOCK LOW(A) TOTAL
SCRATCH
HIGH(B) 70 9 79
RESISTANCE 16 5 21
LOW(B)
TOTAL 86 14 100
Compute P(A).
Therefore, the probability value of the event A is 0.86.
Part a
The probability that the event disk has high shock resistance is 0.86
Explanation | Hint for next step
Based on the given information, the probability that the event disk has high shock resistance is 0.86. That means it is approximately equal to 86%.
Step 2 of 2
(b)
From the given information,
Let A denote the event that a disk has high shock resistance,
and let B denote the event that a disk has high scratch resistance.
SHOCK HIGH(A) SHOCK LOW(A) TOTAL
SCRATCH
HIGH(B) 70 9 79
SCRATCH
LOW(B) 16 5 21
TOTAL 86 15 100
ComputeP(B/A) = P(A∩B) /P(A) ;P(A) > 0
P(A∩B) =70/100
=0.70
From the part [a], the probability value of the event A is P(A) =0.86 .
Therefore,
P( {B/A} = P(A∩B) /P(A)
= 0.70 /0.86
=0.8140
Part b
The probability that a disk has high scratch resistance given that the disk has high shock resistance is 0.8140
Explanation | Common mistakes
The disk with high scratch resistance is found to be 81.40% with the condition that the disk with high shock resistance is maintained.
Final answer:
Events A (high shock resistance) and B (high scratch resistance) in the context of polycarbonate plastic disks are not independent, as the calculated probabilities P(A ∩ B) and P(A)P(B) do not match.
Explanation:
Whether two events are independent can be determined by checking if the probability of one event occurring does not affect the probability of the other event. Mathematically, two events A and B are independent if and only if-
P(A ∩ B) = P(A)P(B)
Let's calculate this using the data provided from the analysis of 100 disks of polycarbonate plastic.
Total number of disks (n) = 100
Number of disks with high shock resistance (A) = 70 + 9 = 79
Number of disks with high scratch resistance (B) = 70 + 16 = 86
Number of disks with both high shock and scratch resistance (A ∩ B) = 70
The probability of A (P(A)) is thus 79/100, the probability of B (P(B)) is 86/100, and the probability of both A and B occurring (P(A ∩ B)) is 70/100. Now we multiply P(A) and P(B):
P(A)P(B) = (79/100) × (86/100) = 0.6794
However, P(A ∩ B) as observed is-
70/100 = 0.7
Since P(A ∩ B) does not equal P(A)P(B), the events A and B are not independent.
The United Kingdom is about 900km long from north to south. By how many orders of magnitude (i.e. how many times) would you have to scale down a map of the United Kingdom so that it can fit on a sheet of paper 20cm long
Answer:
[tex]45*10^{6}[/tex] orders of magnitude.
Step-by-step explanation:
We know that 900 km is equivalent to 20 cm, because we need to fit the United Kingdom length on a sheet of paper. So:
1 cm in a paper will be 45 km in the real ground.Therefore, if we divide 45 km by 0.00001 km (1 cm) we will have [tex]45*10^{6}[/tex] times to scale down.
I hope it helps you!
anju plots points showing equivalent ratios on the coordinate plane below
The missing value in the table is 15.
Solution:
Ratio of x to y coordinate:
[tex]$\frac{x}{y}= \frac{1}{3}[/tex]
[tex]$\frac{x}{y}= \frac{2}{6}= \frac{1}{3}[/tex]
[tex]$\frac{x}{y}= \frac{8}{24}= \frac{1}{3}[/tex]
So, the ratios are equivalent ratios.
To find the missing value in the table:
Multiply the x-coordinate by 3, we get the y-coordinate.
x-coordinate = 5
y-coordinate = x-coordinate × 3
= 5 × 3
= 15
Hence the missing value in the table is 15.
Answer:
15
Step-by-step explanation:
1 multiplied by 5 equals 5. Do the same with the y-value. 3 multiplied by 5 is 15!
Consider this scatter plot.
(a) How would you characterize the relationship between the hours spent on homework and the test
scores? Explain.
(b) Paul uses the function y = 8x + 42 to model the situation. What score does the model predict for 3 h
of homework?
Answer:
a) For this case we can see on the y axis the test scores and on the x axis the hours of homework. And as we can see we have a proportional relationship, because when the hours of homework increase the test scores increases too. If we fit a lineal model between y and x we will see an slope positive and a correlation coefficient positive based on the data observed.
b) [tex] y = 8x +42[/tex]
And we want to predict the test score for x = 3hr of homework we just need to replace the value of x=3 in the linear model and we got:
[tex] y = 8*3 +42= 24+42=66[/tex]
And that would be our predicted value for 3 h of homework
Step-by-step explanation:
For this case we consider the scatter plot attached to solve the problem.
Part a
For this case we can see on the y axis the test scores and on the x axis the hours of homework. And as we can see we have a proportional relationship, because when the hours of homework increase the test scores increases too. If we fit a lineal model between y and x we will see an slope positive and a correlation coefficient positive based on the data observed.
Part b
Assuming the following linear model for the situation:
[tex] y = 8x +42[/tex]
And we want to predict the test score for x = 3hr of homework we just need to replace the value of x=3 in the linear model and we got:
[tex] y = 8*3 +42= 24+42=66[/tex]
And that would be our predicted value for 3 h of homework
A study was done to compare people's religion with how many days that they have said they have been calm in the past three days. 0 days 1 day 2 days 3 days Protestant 50 27 6 16 Catholic 55 19 2 11 Jewish 84 34 4 18 Other 84 34 2 22 What proportion of Protestants were calm for 2 out of the 3 days? NOTE: Read the question very carefully.
Using it's concept, it is found that the proportion of Protestants that were calm for 2 out of the 3 days is of 0.0606.
What is a proportion?A proportion is a fraction of total amount, and is given by the number of desired outcomes divided by the number of total outcomes.In this problem:
There is a total of 50 + 27 + 6 + 16 = 99 Protestants.Of those, 6 were calm for 2 out of the 3 days.Hence:
[tex]p = \frac{6}{99} = 0.0606[/tex]
The proportion of Protestants that were calm for 2 out of the 3 days is of 0.0606.
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Approximately 6.06% of Protestants were calm for 2 out of the 3 days, calculated by dividing the number of Protestants calm for 2 days (6) by the total surveyed (99).
Explanation:To find the proportion of Protestants who were calm for 2 out of the 3 days, we need to look at the numbers given for Protestant individuals and calculate the ratio of those who were calm for exactly two days to the total number of Protestants surveyed. According to the data provided:
Number of Protestants calm for 0 days: 50Number of Protestants calm for 1 day: 27Number of Protestants calm for 2 days: 6Number of Protestants calm for 3 days: 16The total number of Protestants surveyed is the sum of those calm for 0, 1, 2, and 3 days, which is 50 + 27 + 6 + 16 = 99.
To find the proportion of Protestants calm for 2 days, we divide the number calm for 2 days by the total number of Protestants surveyed:
Proportion = Number calm for 2 days / Total number of Protestants
Proportion = 6 / 99
Proportion ≈ 0.0606
This means that approximately 6.06% of Protestants were calm for 2 out of the 3 days.
A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinationsShort-sleeved Pattern Size Pl Pr StS 0.04 0.02 0.05M 0.07 0.10 0.12L 0.03 0.07 0.08Long-sleeved Pattern Size Pl Pr StS 0.03 0.02 0.03M 0.06 0.07 0.07L 0.04 0.02 0.08(a) What is the probability that the next shirt sold is a medium, long-sleeved, print shirt?(b) What is the probability that the next shirt sold is a medium print shirt?(c) What is the probability that the next shirt sold is a short-sleeved shirt?What is the probability that the next shirt sold is a long-sleeved shirt?(d) What is the probability that the size of the next shirt sold is a medium?What is the probability that the pattern of the next shirt sold is a print?(e) Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium?(f) Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? What is the probability that it was long-sleeved?
Answer and Step-by-step explanation:
a) probability of selling a medium, long sleeved printed shirt, P(M ∩LS ∩PR) = 0.07, directly from the table of probabilities
b) probability that the next shirt sold is a medium printed shirt, (M ∩Pr) = P(M,Pr,LS) + P(M,Pr,SS) = 0.07+0.10 = 0.17
c) probability that the next shirt is a short sleeved shirt, P(SS) = sum of 9 probabilities in Short Sleeved shirt table = 0.58
probability that the next shirt is a long sleeved shirt, P(LS) = 1 - 0.58 = 0.42 or add up the total probabilities in the long sleeved shirt table, P(LS) = sum of 9 probabilities in the long sleeved shirt table = 0.42
d) probability that the next shirt is a medium, P(M) = P(M,SS) + P(M,LS) = (0.07+0.10+0.12) + (0.06+0.07+0.07) = 0.49
probability that the next shirt sold is a print, P(Pr) = P(Pr,SS) + P(Pr,LS) = (0.02+0.10+0.07) + (0.02+0.07+0.02) = 0.40
e) probability that the shirt sold is a medium given that the shirt just sold was a short-sleeved plaid, P(M|SS,PL) = (P(M,SS,PL))/P(SS,PL) = 0.07/(0.04+0.07+0.03) = 0.5
f) probability that the shirt sold is short sleeved given that the shirt just sold was a medium plaid, P(SS|M,PL) = (P(M,SS,PL))/P(M,PL) = 0.07/(0.07+0.06) = 0.53846 = 0.54
probability that the shirt sold is long sleeved given that the shirt just sold was a medium plaid, P(LS|M,PL) = (P(M,LS,PL))/P(M,PL) = 0.06/(0.07+0.06) = 0.462 = 0.46
QED!
The following probabilities are as follow;
The probability of a medium, long-sleeved, print shirt is 0.07.The probability of a medium print shirt is 0.17.The probability of a long-sleeved shirt is 0.42.The probability of a medium print shirt is 0.17The probability of a short-sleeved plaid medium is 0.07.The probability of a medium plaid short-sleeved is 0.07.What is probability?Probability means possibility. It deals with the occurrence of a random event. The value of probability can only be from 0 to 1. Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.
(a) The next shirt sold is a medium, long-sleeved, print shirt.
Probability = 0.07
(b) The probability that the next shirt sold is a medium print shirt.
Probability = 0.7 + 0.10 = 0.17
(c) The probability that the next shirt sold is a long-sleeved shirt.
Probability = 0.42
(d) The probability that the size of the next shirt sold is medium. The probability that the pattern of the next shirt sold is a print.
Probability = 0.07 + 0.10 = 0.17
(e)The shirt just sold was a short-sleeved plaid, the probability that its size was medium.
Probability = 0.07
(f) The shirt just sold was a medium plaid, The probability that it was short-sleeved.
Probability = 0.07
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At what points does the helix r(t) = sin(t), cos(t), t intersect the sphere x2 + y2 + z2 = 17? (Round your answers to three decimal places. If an answer does not exist, enter DNE.)
Answer:
the helix intersects the sphere at t=4 and t=(-4)
Step-by-step explanation:
for the helix r(t) = [ sin(t) , cos(t) , t ] then x=sin(t) , y=cos(t) and z=t
thus the helix intersect the sphere x² + y² + z² = 17 at
x² + y² + z² = 17
[sin(t)]²+[cos(t)]²+ t² = 17
1 + t² = 17
t² = 16
t = ±4
thus the helix intersects the sphere at t=4 and t=(-4)
The point wher the helix intersect the sphere is at t = ±4
The coordinate of a helixGiven the coordinate of a helix expressed as;
r(t) = [sin(t), cos(t), t]If these coordinate intersects the sphere x² + y² + z² = 17
Substitute the coordinate of the helix:
x² + y² + z² = 17
(sint)² + (cost)² + t² = 17
sin²t + cos²t + t² = 17
1 + t² = 17
t² = 17 - 1
t² = 16
t = ±√16
t = ±4
Hence the point wher the helix intersect the sphere is at t = ±4
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1. Rewrite each condition below in valid Java syntax (give a boolean expression): a. x > y > z b. x and y are both less than 0 c. neither x nor y is less than 0 d. x is equal to y but not equal to z
Answer:
Here are Boolean expressions in Java syntax.
Step-by-step explanation:
For each part:
a) (x > y && y > z)
b) x == y && x < 0, or x < 0 && y < 0, or x == y && y < 0 (which is essentially the first example)
c) (x == y && x >= 0), or (x >= 0 && y >= 0), or (x == y && y >= 0) (for the first and third, once the first condition establishes that x is equal to y, if either x or y is greater than or equal to 0, then they are both not less than 0)
d) (x == y && x != z), or (x == y && y != z)
An air traffic controller has noted that it clears an average of seven planes per hour for landing. What is the probability that during the next two hours exactly 15 planes will be cleared for landing?a. 0.0989 b. Not enough information is given to answer the problem. c. 0.0033 d. 0.0651
Answer:
a) 0.0989, Option a
Step-by-step explanation:
The concept of Poisson probability distribution is used as shown in the attached file.
A trough has a trapezoidal cross section with a height of 5 m and horizontal sides of width 5/2 m and 5 m. Assume the length of the trough is 10 m. Complete parts? (a) and? (b) below.
a) How much work is required to pump the water out of the trough? (to the level of the top of the? trough) when it is? full? Use 1000 kg/m3 for the density of water and9.8 m/s2 for the acceleration due to gravity. Draw a? y-axis in the vertical direction? (parallel to? gravity) and use the midpoint of the bottom of one edge of the trough as the location of the origin. For 0?y?5?, find the? cross-sectional area? A(y) in terms of y.
b) Set Up the integral that gives the work required to pump the water out of the tank.
Answer:
Step-by-step explanation:
Considering Volume of water inside this = Base area x HeightBase is trapezoid = 0.5 (2.5 + 5) X 5 = 18.75 m2Height = 10 mVolume = 10 * 18.75 = 187.5 m3Total mass of water = density X volume = 1000 kg/m3 x 187.5 m3
= 187500kgWork done to pump out the water = change in potential energy of system of waterWork = mg(H2 - H1)where H2 = initial position of centre of mass from ground
where H1 = final position of centre of mass from ground
here H2 = 2.78 m (from bottom) H1 = 0Work = 187500kg X 9.8 X (2.78 - 0) = 5.11 x 106 JThe detailed analysis of the (b) part is as shown in the attachmentThe work required to pump the water out of the trough is [tex]5.11\times 10^6[/tex] J and this can be determine by using the formula of work done.
Given :
A trough has a trapezoidal cross section with a height of 5 m and horizontal sides of width 5/2 m and 5 m. Assume the length of the trough is 10 m.a) The work required to pump the water out of the trough is given by:
[tex]\rm W=mg(H_2-H_1)[/tex] --- (1)
where m is the total mass of the water [tex]\rm H_1[/tex] is the initial position and [tex]\rm H_2[/tex] is the final position of the center of mass.
So. the mass of the water is given below:
[tex]=1000\times 187.5[/tex]
= 187500 Kg
Now, the value of the initial position of the center of mass is zero and the value of the final position of the center of mass is 2.78 m.
Now, substitute the known terms in the expression (1).
[tex]\rm W = 187500\times 9.8\times (2.78-0)[/tex]
[tex]\rm W = 5.11\times 10^{6}\;J[/tex]
b) The width of the cylinder is given by:
[tex]\rm w-5=\dfrac{5-2.5}{0-5}(x-0)[/tex]
Simplify the above expression.
w = -0.5x + 5
Now, the volume of the cylinder is given by:
[tex]\rm dv = 10wdx[/tex]
Substitute the value of w in the above expression.
[tex]\rm dv = 10(-0.5x+5)dx[/tex]
dv = (50-5x) dx
Now, the value of the force is given by:
[tex]\rm dF = 9800dv[/tex]
Substitute the value of dv in the above expression.
dF = 9800(50 - 5x) dx
Now, the expression of work is given by:
dW = x dF
Substitute the value of force in the above expression.
dW = x(9800(50 - 5x)) dx
Now, integrate the above expression.
[tex]\rm W = 9800\int^5_0(50x-5x^2)dx[/tex]
[tex]\rm W = 9800 \times \left(25x^2-\dfrac{5x^3}{3}\right)^5_0[/tex]
[tex]\rm W= 9800\left(625-\dfrac{625}{3} \right)[/tex]
W = 4083333.34 J
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Suppose that diameters of a new species of apple have a bell-shaped distribution with a mean of 7.46cm and a standard deviation of 0.39cm. Using the empirical rule, what percentage of the apples have diameters that are greater than 7.07cm? Please do not round your answer.
Using the empirical rule, approximately 84% of apple diameters are greater than 7.07 cm, since this value is within one standard deviation below the mean diameter of 7.46 cm.
Explanation:The empirical rule (also known as the 68-95-99.7 rule) is a statistical rule which states that for a normally distributed set of data, approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
In this particular problem, we know the mean (7.46cm) and standard deviation (0.39cm) of the apple diameters. We are asked to find the percentage of apple diameters that are greater than 7.07cm. Since 7.07cm is less than one standard deviation below the mean (7.46 - 0.39 = 7.07), from the normal distribution we can say that approximately 84% (50% of the data is above the mean and 34% is between the mean and one standard deviation below the mean) of the apple diameters are larger than 7.07cm according to the Empirical rule.
Thus, using Empirical rule and given statistical mean and standard deviation, we can estimate that about 84% of the apples from this species have a diameter greater than 7.07 cm.
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Using the Empirical Rule, approximately 84% of the apples have diameters greater than 7.07cm since 7.07cm is exactly one standard deviation below the mean of 7.46cm in a normal distribution.
To answer the student's question, we need to apply the Empirical Rule which is used in statistics to describe the distribution of data in a bell-shaped curve, also known as a normal distribution. This rule states that for a data set with a normal distribution, approximately 68% of the data falls within one standard deviation from the mean, 95% falls within two standard deviations, and over 99% falls within three standard deviations.
Given that the mean diameter of the apples is 7.46cm and the standard deviation is 0.39cm, first we need to determine how many standard deviations 7.07cm is from the mean. This is calculated as:
(7.07cm - 7.46cm) / 0.39cm = -1 standard deviation.
According to the Empirical Rule, 68% of apples are within one standard deviation above and below the mean. Since 7.07cm falls exactly at one standard deviation below the mean, we have:
50% of the apples have diameters greater than the mean.Another 34% (half of 68%) have diameters that fall between the mean and one standard deviation below the mean.Therefore, to find the percentage of apples that have diameters greater than 7.07cm, we would add these two percentages together:
50% + 34% = 84%
Thus, we conclude that approximately 84% of the apples have diameters that are greater than 7.07cm.
Prior Knowledge Questions (Do these BEFORE using the Gizmo.) Imagine you and your friends are making hot dogs. A complete hot dog consists of a wiener and a bun. At the store, you buy four packages of eight wieners and three bags of 10 buns. 1. How many total hot dogs can you make? ________________________________________ 2. Which ingredient limited the number of hot dogs you could make? ____________________ 3. Which ingredient will you have leftovers of? ______________________________________
Answer:
A. 8 complete hotdogs were made.
B. The wiener.
C. The bun.
Step-by-step explanation:
1 complete hotdog = 1 wiener + 1 bun
You have, 3*10 bun and 8 wiener
= 30 bun and 8 wiener
A. Since, we have 30 bun and 8 wiener and for 1 complete hotdog = 1 wiener + 1 bun
Therefore, 8 hotdogs = 8 wiener + 8 buns
= 8 complete hotdogs are made
B.
Since 8 complete hotdogs are made, 8 wiener and 8 buns are used.
Therefore, the limiting ingredient is the wiener because it is the smallest number of ingredient and none of it was left after making the 8 hotdogs.
C. Since 8 complete hotdogs are made, 8 buns were used
So, the leftover buns = (30 - 8) buns
= 22 buns
The buns is the leftover ingredient
At the store, you buy four packages of eight wieners and three bags of 10 buns, you can make 8 hotdogs.
There are 8 hotdogs you can make then the limited ingredient is wieners.
The buns are the leftover ingredient.
Given that,You and your friends are making hot dogs.
A complete hot dog consists of a wiener and a bun.
At the store, you buy four packages of eight wieners and three bags of 10 buns.
According to the question,Total number of buns = 30
Total number of wieners = 8
At the store, you buy four packages of eight wieners and three bags of 10 buns.
For every hot dog, 1 wiener one 1 buns are required. the calculation for the given question is given as follows;
1. The total number of hot dogs can you make is,
To make 1 hot dog 1 wiener and 1 bun
Then,
for 8 hot dogs, 8 wieners and 8 buns are required.
Therefore, you can make 8 hotdogs.
2. There are 8 hotdogs you can make then the limited ingredient is wieners.
3. After making 8 hot dogs only 8 buns are used,
Then,
Left buns = Total number of buns - used buns
Left buns = 30-8 = 22
The buns are the leftover ingredient.
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Suppose you choose a team of two people from a group of n > 1 people, and your opponent does the same (your choices are allowed to overlap). Show that the number of possible choices of your team and the opponent’s team equals Pn−1 i=1 i 3 .
Answer:
The number of possible choices of my team and the opponents team is
[tex]\left\begin{array}{ccc}n-1\\E\\n=1\end{array}\right i^{3}[/tex]
Step-by-step explanation:
selecting the first team from n people we have [tex]\left(\begin{array}{ccc}n\\1\\\end{array}\right) = n[/tex] possibility and choosing second team from the rest of n-1 people we have [tex]\left(\begin{array}{ccc}n-1\\1\\\end{array}\right) = n-1[/tex]
As { A, B} = {B , A}
Therefore, the total possibility is [tex]\frac{n(n-1)}{2}[/tex]
Since our choices are allowed to overlap, the second team is [tex]\frac{n(n-1)}{2}[/tex]
Possibility of choosing both teams will be
[tex]\frac{n(n-1)}{2} * \frac{n(n-1)}{2} \\\\= [\frac{n(n-1)}{2}] ^{2}[/tex]
We now have the formula
1³ + 2³ + ........... + n³ =[tex][\frac{n(n+1)}{2}] ^{2}[/tex]
1³ + 2³ + ............ + (n-1)³ = [tex][x^{2} \frac{n(n-1)}{2}] ^{2}[/tex]
=[tex]\left[\begin{array}{ccc}n-1\\E\\i=1\end{array}\right] = [\frac{n(n-1)}{2}]^{3}[/tex]
Suppose your statistics instructor gave six examinations during the semester. You received the following grades (percent correct): 79, 64, 84, 82,92, and 77. Instead of averaging the six scores, the instructor indicated he would randomly select two grades and report that grade to the student records office.
a. How many different samples of two test grades are possible?
b. List all possible samples of size two and compute the mean of each.
c. Compute the mean of the sample means and compare it to the population mean.
Answer:
a. 15
b.
Sr.no Samples Sample mean
1 (79,64) 71.5
2 (79,84) 81.5
3 (79,82) 80.5
4 (79,92) 85.5
5 (79,77) 78
6 (64,84) 74
7 (64,82) 73
8 (64,92) 78
9 (64,77) 70.5
10 (84,82) 83
11 (84,92) 88
12 (84,77) 80.5
13 (82,92) 87
14 (82,77) 79.5
15 (92,77) 84.5
c.
mean of sample mean=population mean=79.67
Step-by-step explanation:
a.
The different samples of two test grade are nCr, where n=6 and r=2.
nCr=6C2=6!/2!(6-2)!=6*5*4!/2!4!=30/2=15.
Thus, there are 15 different samples of two test grade.
b.
All possible samples are listed below:
Sr.no Samples
1 (79,64)
2 (79,84)
3 (79,82)
4 (79,92)
5 (79,77)
6 (64,84)
7 (64,82)
8 (64,92)
9 (64,77)
10 (84,82)
11 (84,92)
12 (84,77)
13 (82,92)
14 (82,77)
15 (92,77)
The sample means for each sample can be calculated as
Sr.no Samples Sample mean
1 (79,64) (79+64)/2=71.5
2 (79,84) (79+84)/2=81.5
3 (79,82) (79+82)/2=80.5
4 (79,92) (79+92)/2=85.5
5 (79,77) (79+77)/2=78
6 (64,84) (64+84)/2=74
7 (64,82) (64+82)/2=73
8 (64,92) (64+92)/2=78
9 (64,77) (64+77)/2=70.5
10 (84,82) (84+82)/2=83
11 (84,92) (84+92)/2=88
12 (84,77) (84+77)/2=80.5
13 (82,92) (82+92)/2=87
14 (82,77) (82+77)/2=79.5
15 (92,77) (92+77)/2=84.5
c.
The sample means of sample mean μxbar will calculated by taking average of sample means
μxbar=(71.5+ 81.5+ 80.5+ 85.5+ 78+ 74+ 73+ 78+ 70.5+ 83+ 88+ 80.5+ 87+ 79.5+ 84.5)/15
μxbar=1195/15=79.67
Population mean=μ=(79+64+84+82+92+77)/6
μ=478/6=79.67
Sample means of sample mean μxbar=Population mean μ.
You are given the probability that an event will not happen. Find the probability that the event will happen.
1. P(E') = 0.14
2. P(E') = 0.92
3. P(E') = 17/35
4. P(E') = 61/100
Answer:
1) P(E) = 1 - 0.14 = 0.86
2) P(E) = 1 - 0.92 = 0.08
3) P(E) = 1 - 17/35 = 18/35
4) P(E) = 1 - 61/100 = 39/100
Step-by-step explanation:
The probability that an event will happen equals one minus the probability that an event will not happen.
P(E) = 1 - P(E') ......1
Where;
P(E) is the probability that an event will happen.
P(E') is the probability that an event will not happen.
1. P(E') = 0.14
Applying equation 1
P(E) = 1 - 0.14 = 0.86
2. P(E') = 0.92
P(E) = 1 - 0.92 = 0.08
3. P(E') = 17/35
P(E) = 1 - 17/35 = 18/35
4. P(E') = 61/100
P(E) = 1 - 61/100 = 39/100
Farmer Bob's dairy farm is ready for milking day. His 199 cows line up in the barn and he begins milking them. The first cow gives 4 pints of milk. The next cow gives 5 pints of milk. The next gives 6 pints, and so on, increasing by 1 pint each cow. How many pints of milk does the last cow give?
the last cow gives 202 pints of milk
Given :
There are 199 cows line up in the barn and he begins milking them.
The first cow gives 4 pints of milk. The next cow gives 5 pints of milk. The next gives 6 pints, and so on, increasing by 1 pint each cow.
pints of milk is increasing 1 by a sequence
4,5,6,7 ......... and so on
when n=1 , the cow given 4 pints of milk
The sequence is arithmetic
nth cow is 199. Lets find out pints of milk when n=199
apply nth formula of arithmetic
[tex]T_n = T_1 + (n-1) d\\[/tex]
n=199
difference d=1
T1=4
Substitute all the values
[tex]T_n = T_1 + (n-1) d\\\\\\T_{199}=4+(199-1)1\\T_{199}=4+198\\T_{199}=202[/tex]
So, the last cow gives 202 pints of milk
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Rewrite each of the following statements in the form "∀ _____ x, _____." (a) All dinosaurs are extinct. ∀ x, . (b) Every real number is positive, negative, or zero. ∀ x, . (c) No irrational numbers are integers. ∀ x, .(d) No logicians are lazy. ∀ x, .(e) The number 2,147,581,953 is not equal to the square of any integer. ∀ x, .(f) The number −1 is not equal to the square of any real number.
Answer:
See below
Step-by-step explanation:
Essentially, we have to replace "quantifier words" like "All", "Every" by the universal quantifier ∀.
a) ∀ dinosaur x, x is extinct.
b) ∀ real number x, x is positive, negative, or zero.
c) ∀ irrational number x, x is not an integer.
d) ∀ logician x, x is not lazy.
e) ∀ integer x, x²≠ 2,147,581,953.
f) ∀ real number x, x²≠ -1.
In a) and b) we replace the words without major changes. In the other statements, we modify the statement using negation. For example, "No irrational numbers are integers." is equivalent to "Every irrational number is not integer".
A frequent flyer was interested in the relationship between dollars spent on flying and the distance flown. She sampled 20 frequent flyers of a certain airline. She collected the number of miles flown in the previous year and the total amount of money the flyer spent. A regression line of distance flown on money spent was fit to the data, and the intercept and slope were calculated to be a = 24,000 and b = 10.
A person who spent $ 2000 is predicted to have flown:
a) 34000 miles. b) 54000 miles. c) 44000 miles. d) 24000 miles.
Answer:
c) 44000 miles
Step-by-step explanation:
The regression line has the following format:
[tex]y = bx + a[/tex]
In which b is the slope(how much each mile costs) and a is the fixed number of miles flown.
In this problem, we have that:
[tex]a = 24000, b = 10[/tex]. So
[tex]y = 10x + 24000[/tex]
A person who spent $ 2000 is predicted to have flown:
This is y when x = 2000.
[tex]y = 10(2000) + 24000[/tex]
[tex]y = 44000[/tex]
So the correct answer is:
c) 44000 miles
To find the predicted distance flown by someone who spent $2000, you use the given intercept (a) and slope (b) in the straight line equation Y = a + bX. When $2000 is substituted for X in the equation, your solution is 44,000 miles.
Explanation:In this problem, you are asked to determine the distance a person is predicted to have flown, given that the person spent $2000. Based on our regression line information, we know that the intercept (a) is 24,000 and the slope (b) is 10. Therefore, we can use the equation of the straight line Y = a + bX, where X represents the amount of money spent, and Y is the distance traveled. If we substitute $2000 for X in our equation, we find that Y = 24000 + 10 * 2000 = 44,000 miles. Therefore, the correct answer is option c) 44000 miles.
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How many ways are there to pick a group of n people from 100 people (each of a different height) and then pick a second group of m other people such that all people in the first group are taller than the people in the second group
The total number of ways to choose the two groups, depending on the relationship between n and m, is:
M < N: C(100, n).
M ≥ N: Σ (C(100, k) * C(100 - k, m - k)) for k = n to 1.
Case 1: m < n (n tallest chosen first):
Choose the n tallest people from the 100. There are C(100, n) ways to do this.
The remaining 100 - n people are all shorter than the chosen n. Since m < n, we can simply choose all the remaining people (100 - n).
The total number of ways in this case is C(100, n).
Case 2: m ≥ n (not all n tallest need to be chosen):
This case requires considering all possible combinations of how many tall people to choose (from n down to 1).
For each value of k (n, n - 1, ..., 1), choose k tallest people from the 100. There are C(100, k) ways to do this.
The remaining 100 - k people are all shorter than the chosen k. Choose the remaining m - k people from this group. There are C(100 - k, m - k) ways to do this.
Sum the results for each value of k: Σ (C(100, k) * C(100 - k, m - k)) for k = n to 1.
Therefore, the total number of ways to choose the two groups, depending on the relationship between n and m, is:
M < N: C(100, n)
M ≥ N: Σ (C(100, k) * C(100 - k, m - k)) for k = n to 1
Complete question:
How many ways are there to pick a group of n people from 100 people, and then pick a second group of m people from the remaining 100 − n people, so that all the people you chose from the first group are taller than the people from the second group (assume that everyone in the group of 100 people has a different height.)
Note: it is implied in this question that 1 ≤ n ≤ 100 and 0 ≤ m ≤ 100 − n.
If the work required to stretch a spring 2 ft beyond its natural length is 6 ft-lb, how much work is needed to stretch it 6 in. beyond its natural length?
Answer:
0.375 feet-lb
Step-by-step explanation:
We have been given that the work required to stretch a spring 2 ft beyond its natural length is 6 ft-lb. We are asked to find the work needed to stretch the spring 6 in. beyond its natural length.
We can represent our given information as:
[tex]6=\int\limits^2_0 {F(x)} \, dx[/tex]
We will use Hooke's Law to solve our given problem.
[tex]F(x)=kx[/tex]
Substituting this value in our integral, we will get:
[tex]6=\int\limits^2_0 {kx} \, dx[/tex]
Using power rule, we will get:
[tex]6=\left[ \frac{kx^2}{2} \right ]^2_0[/tex]
[tex]6=\frac{k(2)^2}{2}-\frac{k(0)^2}{2}[/tex]
[tex]6=\frac{4k}{2}-0\\\\k=3[/tex]
We know that 6 inches is equal to 0.5 feet.
Work needed to stretch it beyond 6 inches beyond its natural length would be [tex]\int\limits^{0.5}_0 {kx} \, dx =\int\limits^{0.5}_0 {3x} \, dx[/tex]
Using power rule, we will get:
[tex]\int\limits^{0.5}_0 {3x} \, dx = \left [\frac{3x^2}{2}\right]^{0.5}_0[/tex]
[tex]\frac{3(0.5)^2}{2}-\frac{3(0)^2}{2}\Rightarrow \frac{3(0.25)}{2}-0=\frac{0.75}{2}=0.375[/tex]
Therefore, 0.375 feet-lb work is needed to stretch it 6 in. beyond its natural length.
The revenue (in millions of dollars) from the sale of x units at a home supply outlet is given by R(x) = .21x. The profit (in millions of dollars) from the sale of x units is given by P(x) = .084x - 1.5. a. Find the cost equation. b. What is the cost of producing 7 units? c. What is the break-even point?
Answer: a) C = 0.126x + 1.5
b) C = 2.382 c) approximately 179 units
Step-by-step explanation: profit P(x), total revenue R(x) and total cost (C) are related by the formulae below.
Profit = total revenue - total cost
P(x) = R(x) - C
From the question, P(x) = 0.084x - 1.5, R(x) = 0.21x.
Question a)
C = R(x) - P(x)
C = 0.21x - {0.084x - 1.5}
C = 0.21x - 0.084x + 1.5
C = 0.126x + 1.5
Question b)
If C = 0.126x + 1.5, then C at x = 7 units is gotten below as
C = 0.126(7) + 1.5
C = 0.882 + 1.5
C = 2.382.
Question c)
The break even point is the point where total revenue R(x) equals total cost C.
R(x) = 0.21x and C = 0.126x + 1.5
0.21x = 0.126x + 1.5
0.21x - 0.126x = 1.5
0.084x = 15
x = 15/ 0.084
x = 178.57 which is approximately 179 units (since quantity of units can't be decimal)
To find the cost equation, subtract the profit equation from the revenue equation. The cost of producing 7 units is approximately 2.382 million dollars. The break-even point occurs at approximately 17.857 units.
Explanation:a. To find the cost equation, we need to subtract the profit equation from the revenue equation. Since the revenue equation is R(x) = .21x and the profit equation is P(x) = .084x - 1.5, the cost equation is C(x) = R(x) - P(x). Substituting the given equations, we have C(x) = .21x - (.084x - 1.5).
Simplifying the equation, we get C(x) = .21x - .084x + 1.5.
Combining like terms, the cost equation is C(x) = .126x + 1.5.
b.
To find the cost of producing 7 units, we can substitute x = 7 into the cost equation. C(7) = .126(7) + 1.5 = .882 + 1.5 = 2.382 million dollars.
c.
The break-even point occurs when the revenue equals the cost. To find the break-even point, we set R(x) = C(x). Substituting the given equations, we have .21x = .126x + 1.5.
Solving for x, we get .084x = 1.5.
Dividing both sides by .084, we find x = 17.857 units.
Therefore, the break-even point is approximately 17.857 units.
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