Answer:
N = 6300 ways
She can choose the materials 6300 ways
Step-by-step explanation:
In this case order of selection is not important, so we use combination.
For solids,
She needs 2 out of 9 available solids = 9C2
For striped prints
She needs 4 out of 7 available = 7C4
For floral prints
She needs 4 out of 5 available = 5C4
The total number of ways she can choose the materials is;
N = 9C2 × 7C4 × 5C4
N = 9/(7!2!) × 7/(4!3!) × 5/(4!1!)
N = 6300 ways
To find the number of different ways Courtney can choose the materials for her quilt, we can use the concept of combinations. The total number of ways she can choose the materials is the product of the number of choices for each type of fabric.
Explanation:To find the number of different ways Courtney can choose the materials for her quilt, we can use the concept of combinations. The total number of ways she can choose the materials is given by the product of the number of choices for each type of fabric. So, the answer is:
Total number of ways = number of ways to choose solids * number of ways to choose floral prints * number of ways to choose striped fabrics
Given that she needs 2 solids, 4 floral prints, and 4 striped fabrics, we can calculate:
Number of ways to choose solids = combinations(9, 2) = 36Number of ways to choose floral prints = combinations(5, 4) = 5Number of ways to choose striped fabrics = combinations(7, 4) = 35Substituting these values into the formula:
Total number of ways = 36 * 5 * 35 = 6300
So, there are 6300 different ways Courtney can choose the materials for her quilt.
Kelly plan to fence in her yard. The fabulous fence company charges $3.25 per foot of fencing and $15.57 an hour for labor. If Kelly needs 350 feet of fencing and the installers work a total of 6 hour installing the fence , how
much will she owe the fabulous fence company.
Answer:
Kelly will owe $1320.92 to the fabulous fence company.
Step-by-step explanation:
There is a cost related to the number of hours and a cost per feet. So the total cost is:
[tex]T = C_{h} + C_{f}[/tex]
In which [tex]C_{h}[/tex] is the cost related to the number of hours and [tex]C_{f}[/tex] is the cost related to the number of feet.
Cost per hour
Each hour costs $15.57.
They work for 6 hours total. So
[tex]C_{h} = 15.57*6 = 93.42[/tex]
Cost per feet
Each feet costs $3.25.
Kelly needs 350 feet. So
[tex]C_{f} = 350*3.25 = 1137.5[/tex]
The total cost is:
[tex]T = C_{h} + C_{f} = 93.42 + 1137.5 = 1230.92[/tex]
Kelly will owe $1320.92 to the fabulous fence company.
Five players agree to divide a cake fairly using the last diminisher method. The players play in the following order: Anne first, Betty second, Cindy third, Doris fourth, and Ellen last. In round 1, there are no diminishers In round 2, Doris is the only diminisher In round 3, Cindy and Ellen are the only diminishers Which player gets her fair share at the end of:
Using the Last Diminisher method, in the first round, Anne gets her fair share because no one diminishes. In the second round, Doris is the only one who diminishes, thus gets her fair share. In the third round, despite Cindy and Ellen both diminishing, Ellen gets her fair share because she is later in turn order.
Explanation:The Last Diminisher method is a fair division protocol used when a divisible good, like a cake in this example, needs to be divided amongst several players. This method removes discrepancies by having each player in turn reduce the piece until they don't want to diminish it further, and then giving that piece to the last to diminish.
In this case, Anne, Betty, Cindy, Doris, and Ellen are dividing the cake and playing in that order. In the first round, no one diminishes, so Anne gets her fair share of the cake. In the second round, Doris is the only one who diminishes, so she gets her fair share at the end of this round. In the third round, the last to diminish are Cindy and Ellen, but since Ellen is later in order, Ellen is the one who gets her fair share at the end of the round.
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A car rental company charges a one-time application fee of 40 dollars, 55 dollars per day, and 13 cents per mile for its cars. A) Write a formula for the cost, C, of renting a car as a function of the number of days, d, and the number of miles driven, m. B) If C = f(d, m), then f(5, 600) =
Answer:
A) C(d,m) = 40 + 55d + 0.13m
B) $448
Step-by-step explanation:
Let 'd' be the number of days and 'm' the number of miles driven.
A) The cost function that describes a fixed amount of $40, added to a variable amount of $55 per day (55d) and a variable amount of 13 cents per mile (0.13m) is:
[tex]C(d,m) = 40 +55d +0.13m[/tex]
B) If d = 5 and m =600, the total cost is:
[tex]C(5,600) = 40 +55*6 +0.13*600\\C(5,600)=\$448[/tex]
The cost is $448.
The formula for the cost, C of renting a car as a function of the number of days, d, and the number of miles driven, m is C f(d, m) = 40 + 55d + 0.13m
Given:
Application fee = $40
cost per day = $55
cost per mile = $0.13
let
Total cost = C
Number of days = d
Number of miles = m
Total cost, C = 40 + 55d + 0.13m
C f(d, m) = 40 + 55d + 0.13m
= 40 + 55(5) + 0.13(600)
= 40 + 275 + 78
= $393
Therefore, the cost of renting the car given the number of days is $393
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You go to Applebee’s and spend $98.42 on your meal. How much was the bill before 6% sales tax
Answer: the bill before 6% sales tax is $92.85
Step-by-step explanation:
Let x represent the bill before the 6% sales tax.
It means that you paid 6% tax on x and the amount of tax paid would be
6/100 × x = 0.06 × x = 0.06x
Total amount that you paid for the meal including the 6% tax would be
x + 0.06x = 1.06x
If you spent $98.42 on the meal after the 6% tax, it means that
1.06x = 98.42
Dividing the left hand side and the right hand side of the equation by 1.06, it becomes
1.06x/1.06 = 98.42/1.06
x = $92.85
A copyeditor thinks the standard deviation for the number of pages in a romance novel is six. A sample of 25 novels has a standard deviation of nine pages. At , is this higher than the editor hypothesized?
Answer:
No, the standard deviation for number of pages in a romance novel is six only.
Step-by-step explanation:
First we state our Null Hypothesis, [tex]H_o[/tex] : [tex]\sigma[/tex] = 6
and Alternate Hypothesis, [tex]H_1[/tex] : [tex]\sigma[/tex] > 6
We have taken these hypothesis because we have to check whether our population standard deviation is higher than what editor hypothesized of 6 pages in a romance novel.
Now given sample standard deviation, s = 9 and sample size, n = 25
To test this we use Test Statistics = [tex]\frac{(n-1)s^{2} }{\sigma^{2} }[/tex] follows chi-square with (n-1) degree of freedom [[tex]\chi ^{2}_n__-1[/tex]]
Test Statistics = [tex]\frac{(25-1)9^{2} }{6^{2} }[/tex] follows [tex]\chi ^{2}_2_4[/tex] = 54
and since the level of significance is not stated in question so we assume it to be 5%.
Now Using chi-square table we observe at 5% level of significance the [tex]\chi ^{2}_2_4[/tex] will give value of 36.42 which means if our test statistics will fall below 36.42 we will reject null hypothesis.
Since our Test statistics is more than the critical value i.e.(54>36.42) so we have sufficient evidence to accept null hypothesis and conclude that our population standard deviation is not more than 6 pages which the editor hypothesized.
There are two traffic lights on Darlene's route from home to work. Let E denote the event that Darlene must stop at the first light, and define the event F in a similar manner for the second light. Suppose that P(E) = 0.2, P(F) = 0.3, and P(E ∩ F) = 0.13.
Answer:
The question is incomplete, below is the complete question,"There are two traffic lights on Darlene's route from home to work. Let E denote the event that Darlene must stop at the first light, and define the event F in a similar manner for the second light. Suppose that P(E) = 0.2, P(F) = 0.3, and P(E ∩ F) = 0.13.
a) What is the probability that the individual needn't stop at either light?
b) What is the probability that the individual must stop at exactly one of the two lights? c) What is the probability that the individual must stop just at the first light?"
Answer:
A. 0.63
B. 0.24
C. 0.07
Step-by-step explanation:
Data given,
P(E) = 0.2, P(F) = 0.3, and P(E ∩ F) = 0.13.
From the question, we can conclude that the event are dependent, hence
a. P(needn't stop at either light) = 1 - P(Need to stop at either light)
P(EUF)' =1-P(EUF)
P(EUF)' =1- (P(E)+P(F) -P(E ∩ F))
P(EUF)' =1-(0.2+0.3-0.13)
P(EUF)' =1-0.37
P(EUF)' =0.63
b. P(must stop at exactly one of the two lights) = P(must stop at either light) - P(must stop at both lights)
P(must stop at exactly one of the two lights) = P(E u F) - P(En F)
but P(E u F)=0.37,
P(En F)=0.13,
P(must stop at exactly one of the two lights) = 0.37 - 0.13 = 0.24
c. P(must stop at just the first light) = P(must stop at either light) - P(must stop at the second light)
P(must stop at just the first light) = P(E u F)-P(F)
P(must stop at just the first light) = 0.37 - 0.3 = 0.07
The question deals with the topic of Probability in Mathematics. It presents the probabilities of two events, denoted as E and F, which are stopping at the first and second traffic lights, respectively. The question also provides the concurrent occurrence of both events.
Explanation:The mathematics topic this question deals with is Probability. In the scenario given, E represents the event that Darlene must stop at the first traffic light and F represents the event that she needs to stop at the second traffic light. The probabilities of these events are given as P(E)=0.2 and P(F)=0.3, respectively. Additionally, we're given that the probability of both events happening (denoted P(E ∩ F)) is 0.13.
In order to analyze the situation, we can leverage the rule of joint probability, which states that the probability of two independent events both happening is the product of their individual probabilities. However, in this case the events E and F are not independent (since the probability of the intersection P(E ∩ F) is not equal to the product of probabilities P(E)*P(F)) so we know that the occurrence of E does influence the occurrence of F, and vice versa.
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Using your knowledge of exponential and logarithmic functions and properties, what is the intensity of a fire alarm that has a sound level of 120 decibels?
A.
1.0x10^-12 watts/m^2
B.
1.0x10^0 watts/m^2
C.
12 watts/m^2
D.
1.10x10^2 watts/m^2
Option B:
[tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]
Solution:
Given sound level = 120 decibel
To find the intensity of a fire alarm:
[tex]$\beta=10\log\left(\frac{I}{I_0} \right)[/tex]
where [tex]I_0=1\times10^{-12}\ \text {watts}/ \text m^2}[/tex]
Step 1: First divide the decibel level by 10.
120 ÷ 10 = 12
Step 2: Use that value in the exponent of the ratio with base 10.
[tex]10^{12}[/tex]
Step 3: Use that power of twelve to find the intensity in Watts per square meter.
[tex]$10^{12}=\left(\frac{I}{I_0} \right)[/tex]
[tex]$10^{12}=\left(\frac{I}{1\times10^{-12}\ \text {watts}/ \text m^2} \right)[/tex]
Now, do the cross multiplication,
[tex]I=10^{12}\times1\times\ 10^{-12} \ \text {watts}/ \text m^2}[/tex]
[tex]I=1\times\ 10^{12-12} \ \text {watts}/ \text m^2}[/tex]
[tex]I=1\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]
[tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]
Option B is the correct answer.
Hence [tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex].
Final answer:
To find the intensity of a sound at 120 dB, we use the formula SIL = 10 log(I / I0). With I0 as 10⁻¹² W/m², we find that I = 1.0 x 10⁰ W/m², corresponding to choice B.
Explanation:
To determine the intensity of a fire alarm that has a sound level of 120 decibels (dB), we use the relationship between sound intensity level and intensity in watts per meter squared (W/m²). The formula to convert decibel level to intensity is:
SIL = 10 log(I / I0)
Where SIL is the sound intensity level in decibels, I is the intensity of the sound, and I0 is the reference intensity, usually taken as 10⁻¹² W/m², the threshold of human hearing. To find the unknown intensity I, we can rearrange the formula:
I = I0 × 10(SIL/10)
For a sound level of 120 dB, the calculation would be:
I = 10⁻¹² W/m² × 10¹²⁰/¹⁰
I = 10⁻¹² W/m² × 10¹²
I = 1.0 × 10⁰ W/m²
Therefore, the correct answer is B. 1.0 x 10⁰ watts/m².
Determine if b is a linear combination of a1 a2, and a3. a1 = [ 1 -2 0 ], a2 = [ 0 1 3 ], a3 = [ 6 -6 18 ], b = [ 2 -2 6 ] Choose the correct answer below. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the fourth column. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the third column. Vector b is not a linear combination of a1, a2, and a3. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column and the second entry in the second column.
Answer: Vector b is not a linear combination
Step-by-step explanation:
First of all we put the vectors in terms of different variables, such as:
a1(1,-2,0)=(a,-2a,0);
a2(0,1,3)=(0,b,3b);
a3(6,-6,18)=(6c,-6c,18c);
To know that a vector is a linear combination we need to express it like a sum of other different vectors.
(2,-2,6)=(a,-2a,0)+(0,b,3b)+(6c,-6c,18c)
(2,-2,6)=(a+0+6c,-2a+b-6c,0+3b+18c)
We express this sum like a system of equations.
a+6c=2
-2a+b-6c=-2
3b+18c=6
We solve this system of equations and we can note that the system don't have a solution, so the vector b is not a linear combination of a1, a2, and a3.
Upon forming a system of linear equations and solving, a solution would imply that vector b is indeed a linear combination of vectors a1, a2, and a3. The observed placement of pivots in the corresponding echelon matrix backs this conclusion.
Explanation:In this question, you are asked to determine if vector b is a linear combination of vectors a1, a2, and a3. A vector is a linear combination of others if it can be written as a weighed sum of those vectors. To solve this problem, we need to form a system of linear equations based on the vectors and solve this system. If all of the coefficients can be expressed as real numbers, it means that the vector b is a linear combination of a1, a2, and a3.
In this case, our system of equations looks like this:
x∗a1 + y∗a2 + z∗a3 = b
In matrix form it can be written as:
|1 0 6|x| = |2|, |-2 1 -6|y| = |-2|, |0 3 18|z| = |6|.
Solve this system through methods like Gauss-Jordan elimination or row reduction. The pivots in the corresponding echelon matrix should be in the first entry in the first column, the second entry in the second column, and the third entry in the third column.
This suggests that vector b can indeed be a linear combination of vectors a1, a2, and a3.
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Joe, Megan, and Santana are salespeople. Their sales manager has 21 accounts and must assign seven accounts to each of them. In how many ways can this be done?
Answer:
116,280 ways
Step-by-step explanation:
The number of ways of assigning the accounts to each of the salesperson is computed by combination
Number of ways = n combination r = n!/(n-r)!r!
n = 21, r = 7
Number of ways = 21 combination 7 = 21!/(21-7)!7! = 21!/14!7! = 116,280 ways
A water tank has 1,500 liters of water. It has a leak, losing 4 liters per minute. At the same time, a second tank has 300 liters and is being filled at a rate of 6 liters per second. Make a system of equations. After how many minutes will they have a same amount of water in the tank?
Let [tex]t[/tex] be the number of minutes.
The first tank starts with 1500 liters, and loses 4 liters per minute, so after [tex]t[/tex] minutes there will be
[tex]1500-4t[/tex]
liters of water.
The second tank is filled at 6 liters per second, i.e.
[tex]6\times 60=360[/tex] liters per minute.
So, there will be
[tex]300+360t[/tex]
liters of water in the second tank after [tex]t[/tex] minutes.
The two quantities will be equal when
[tex]1500-4t=300+360t \iff 1200=364t \iff t=\dfrac{1200}{364}\approx 3.3[/tex]
so, approximately, after 3.3 minutes.
Answer: it will take about 3.3 minutes for both tanks to have the same amount of water.
Step-by-step explanation:
Let x represent the number of minutes it will take both tanks to have same amount of water.
A water tank has 1,500 liters of water. It has a leak, losing 4 liters per minute. This means that in x minutes, the volume of water in the tank would be
1500 - 4x
At the same time, a second tank has 300 liters and is being filled at a rate of 6 liters per second. Converting 6 liters per second to minutes, it becomes 60 × 6 = 360 liters per minute. This means that in x minutes, the volume of water in the tank would be
300 + 360x
For both tanks to have same amount of water, then
1500 - 4x = 300 + 360x
360x + 4x = 1500 - 300
364x = 1200
x = 1200/364 = 3.3 minutes
The Insure.com website reports that the mean annual premium for automobile insurance in the United States was $1,503 in March 2014. Being from Pennsylvania at that time, you believed automobile insurance was cheaper there and decided to develop statistical support for your opinion. A sample of 25 automobile insurance policies from the state of Pennsylvania showed a mean annual premium of $1,425 with a standard deviation ofs = $160.(a) Develop a hypothesis test that can be used to determine whether the mean annual premium in Pennsylvania is lower than the national mean annual premium.H0: μ ≥ 1,503Ha: μ < 1,503H0: μ ≤ 1,503Ha: μ > 1,503 H0: μ > 1,503Ha: μ ≤ 1,503H0: μ < 1,503Ha: μ ≥ 1,503H0: μ = 1,503Ha: μ ≠ 1,503(b) What is a point estimate in dollars of the difference between the mean annual premium in Pennsylvania and the national mean? (Use the mean annual premium in Pennsylvania minus the national mean.)
Final answer:
The hypothesis test to determine the mean annual premium in Pennsylvania compared to the national mean annual premium is H0: μ ≥ 1,503 and Ha: μ < 1,503. The point estimate of the difference between the mean annual premiums is -$78.
Explanation:
(a) To determine whether the mean annual premium in Pennsylvania is lower than the national mean annual premium, we need to develop a hypothesis test. The null hypothesis (H0) states that the mean annual premium in Pennsylvania is greater than or equal to the national mean annual premium. The alternative hypothesis (Ha) states that the mean annual premium in Pennsylvania is less than the national mean annual premium. Therefore, the correct answer is:
H0: μ ≥ 1,503
Ha: μ < 1,503
(b) The point estimate in dollars of the difference between the mean annual premium in Pennsylvania and the national mean is calculated by subtracting the national mean annual premium ($1,503) from the mean annual premium in Pennsylvania ($1,425). Therefore, the point estimate is $1,425 - $1,503 = -$78.
A new streaming service charges 5 dollars per month for students, and 10 dollars per month for everyone else. This month, the service had 55 users, and collected 425 dollars. Set up a system of linear equations, and find the number of students using the service this month.
Answer:
Number of student = 25
Step-by-step explanation:
Let x be the number of student and y be the others
A new streaming service charges 5 dollars per month for students, and 10 dollars per month for everyone else
[tex]x+y=55\\y=55-x[/tex]
[tex]5x+10y=425[/tex]
replace y with 55-x
[tex]5x+10y=425\\5x+10(55-x)=425\\5x+550-10x=425\\-5x+550= 425[/tex]
Subtract 550 from both sides
[tex]-5x+550= 425\\-5x= -125\\x=25[/tex]
[tex]y=55-x\\y=55-25\\y=30\\[/tex]
Number of student = 25
Answer: 25 students used the service this month.
Step-by-step explanation:
Let x represent the number of students that used the streaming service this month.
Let y represent the number of people apart from students that used the streaming service this month.
This month, the service had 55 users. It means that
x + y = 55
The new streaming service charges 5 dollars per month for students, and 10 dollars per month for everyone else. They collected a total of 425 dollars. It means that
5x + 10y = 425 - - - - - - -1
Substituting x = 55 - y into equation 1, it becomes
5(55 - y) + 10y = 425
275 - 5y + 10y = 425
- 5y + 10y = 425 - 275
5y = 150
y = 150/5 = 30
x = 55 - y = 55 - 30
x = 25
In a Caesar Cipher system, if encryption is done by the rule c = E(p+k) = p + 9, then the letter ‘q’ is encrypted as what other letter?
Answer:
i
Step-by-step explanation:
There has to be a mod 26 operation for c, since the alphabet only has 26 letters.
But otherwise
[tex]c = E(p+k) = p + 9 \mod{26}[/tex]
p is the initial position of the letter. So for example, a = 0, b = 1, c = 2, q = 16, z = 25...
9 is by how much the encrypted character will change relative to the original.
We want to cipher the letter q.
So p = 25.
[tex]c = E(p+k) = p + 9 \mod{26}[/tex]
[tex]c = 25 + 9 \mod{26}[/tex]
[tex]c = 34 \mod{26}[/tex]
The remainder of 34 divided by 26 is 8.
So
[tex]c = 8[/tex]
Which means that q is encrypted as 'i'.
In a Caesar Cipher system, when the encryption rule is shifting by 9 places, the letter 'q' is encrypted as 'c'.
Explanation:In a Caesar Cipher system, the letter 'q' is encrypted using the formula:
c = E(p + k)
Where:
c is the ciphertext (the encrypted letter).
p is the plaintext (the original letter).
k is the encryption key.
You mentioned that the encryption is done with the rule c = p + 9. In this case, k is 9.
To encrypt the letter 'q', you would substitute p with 'q' in the formula:
c = 'q' + 9
Adding 9 to 'q' in the alphabet results in:
'q' + 9 = 'z'
So, the letter 'q' is encrypted as the letter 'z' using this Caesar Cipher system.
Therefore, the encrypted letter for 'q' in a Caesar Cipher system with a shift of 9 is 'c'.
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The computers of nine engineers at a certain company are to be replaced. Four of the engineers have selected laptops and the other 5 have selected desktops. Suppose that four computers are randomly selected.
(a) How many different ways are there to select four of the eight computers to be set up?
(b) What is the probability that exactly three of the selected computers are desktops?
(c) What is the probability that at least three desktops are selected?
Answer:
(a) There are 70 different ways set up 4 computers out of 8.
(b) The probability that exactly three of the selected computers are desktops is 0.305.
(c) The probability that at least three of the selected computers are desktops is 0.401.
Step-by-step explanation:
Of the 9 new computers 4 are laptops and 5 are desktop.
Let X = a laptop is selected and Y = a desktop is selected.
The probability of selecting a laptop is = [tex]P(Laptop) = p_{X} = \frac{4}{9}[/tex]
The probability of selecting a desktop is = [tex]P(Desktop) = p_{Y} = \frac{5}{9}[/tex]
Then both X and Y follows Binomial distribution.
[tex]X\sim Bin(9, \frac{4}{9})\\ Y\sim Bin(9, \frac{5}{9})[/tex]
The probability function of a binomial distribution is:
[tex]P(U=k)={n\choose k}\times(p)^{k}\times (1-p)^{n-k}[/tex]
(a)
Combination is used to determine the number of ways to select k objects from n distinct objects without replacement.
It is denotes as: [tex]{n\choose k}=\frac{n!}{k!(n-k)!}[/tex]
In this case 4 computers are to selected of 8 to be set up. Since there cannot be replacement, i.e. we cannot set up one computer twice or thrice, use combinations to determine the number of ways to set up 4 computers of 8.
The number of ways to set up 4 computers of 8 is:
[tex]{8\choose 4}=\frac{8!}{4!(8-4)!}\\=\frac{8!}{4!\times 4!} \\=70[/tex]
Thus, there are 70 different ways set up 4 computers out of 8.
(b)
It is provided that 4 computers are randomly selected.
Compute the probability that exactly 3 of the 4 computers selected are desktops as follows:
[tex]P(Y=3)={4\choose 3}\times(\frac{5}{9})^{3}\times (1-\frac{5}{9})^{4-3}\\=4\times\frac{125}{729}\times\frac{4}{9}\\ =0.304832\\\approx0.305[/tex]
Thus, the probability that exactly three of the selected computers are desktops is 0.305.
(c)
Compute the probability that of the 4 computers selected at least 3 are desktops as follows:
[tex]P(Y\geq 3)=1-P(Y<3)\\=1-[P(Y=0)+P(Y=1)+P(Y=2)]\\=1-[({4\choose 0}\times(\frac{5}{9} )^{0}\times (1-\frac{5}{9} )^{4-0}+({4\choose 1}\times(\frac{5}{9} )^{1}\times (1-\frac{5}{9} )^{4-1}+({4\choose 2}\times(\frac{5}{9} )^{2}\times (1-\frac{5}{9} )^{4-2}]\\=1-0.59918\\=0.40082\\\approx0.401[/tex]
Thus, the probability that at least three of the selected computers are desktops is 0.401.
Solve 4x2 - x + 5 = 0.
Answer:
x
=
1
+
i
√
79
8
,
1
−
i
√
79
8
Step-by-step explanation:
A department store surveyed 428 shoppers, and the following information was obtained: 216 shoppers made a purchase, and 294 were satisfied with the service they received. If 47 of those who made a purchase were not satisfied with the service, how many shoppers did the following?a. made a purchase and were satisfied with the service
b. made a purchase or were satisfied with the serice
c. were satisfied with the service but did not mak a purchase
d. were not satisfied and did not make a purchase
The answer are (a) 169 (b) 341 (c) 125 (d) 87
What is a Venn diagram?A Venn diagram is an illustration that uses circles to show the commonalities and differences between things or groups of things.
Given that, A department store surveyed 428 shoppers, and the following information was obtained: 216 shoppers made a purchase, and 294 were satisfied with the service they received. If 47 of those who made a purchase were not satisfied with the service,
Refer to the Venn diagram attached.
The total number of shoppers surveyed is, N = 428.
Number of shoppers who made a purchase, n (P) = 216
Number of shoppers who were satisfied with the service they received,
n (S) = 294
Number of shoppers who made a purchase but were not satisfied with the service, n(S' ∩ P) = 47
(a) The number of shoppers who made a purchase and were satisfied with the service = n(S ∩ P)
n(S ∩ P) = n(P)-n(S'∩P)
= 216 - 47 = 169
(b) The numbers of shoppers who made a purchase or were satisfied with the service = n (P ∪ S)
n (P ∪ S) = n(P)+n(S)-n(S∩P)
= 216+294-169
= 341
(c) The numbers of shoppers who were satisfied with the service but did not make a purchase = n(S∩P')
= n(S)-n(S∩P)
= 241-169
= 125
(d) The number of shoppers who were not satisfied and did not make a purchase = n(S'∩P')
= N-n (S ∪ P)
= 428-341
= 87
Hence, the answer are (a) 169 (b) 341 (c) 125 (d) 87
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a. 169 shoppers made a purchase and were satisfied with the service.
b. 341 shoppers made a purchase or were satisfied with the service.
c. 125 shoppers were satisfied with the service but did not make a purchase.
d. 381 shoppers were not satisfied and did not make a purchase.
Let's break down the information given:
Total shoppers surveyed = 428
Shoppers who made a purchase = 216
Shoppers satisfied with the service = 294
Shoppers who made a purchase and were not satisfied = 47
We are asked to find:
a. Shoppers who made a purchase and were satisfied with the service.
To find this, we subtract the shoppers who made a purchase and were not satisfied from the total shoppers who made a purchase:
216 − 47 = 169
b. Shoppers who made a purchase or were satisfied with the service.
To find this, we add the shoppers who made a purchase and the shoppers who were satisfied, but we need to be careful not to count the overlap twice (those who made a purchase and were satisfied):
216+294−169=341
c. Shoppers who were satisfied with the service but did not make a purchase.
To find this, we subtract the shoppers who made a purchase and were satisfied from the total shoppers who were satisfied:
294−169=125
d. Shoppers who were not satisfied and did not make a purchase.
To find this, we subtract the shoppers who made a purchase and were not satisfied from the total shoppers surveyed:
428−47=381
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The sum of 5 times a number and
minus −2, plus 7 times a number
Answer:
12x + 2
Step-by-step explanation:
Let the number be represented by x.
Then five times the number = 5*x
Seven times the number = 7*x
Sum of 5 times the number minus -2 = [tex]\[5*x - (-2)\][/tex] = [tex]\[5x +2\][/tex]
Adding seven times the number to this expression yields, [tex]\[5x+2+7x\][/tex]
[tex]\[= (5+7)x+2\][/tex]
[tex]\[= 12x+2\][/tex]
So the simplified expression corresponds to 12x + 2.
Trying to find volume of a right circular cone.
Volume of the cone is 117.5 π ft³
Step-by-step explanation:
Lateral area of the cone = πrs
s is the slant height = 15 ft
From the above formula, we can find the radius as, 5 ft.
Volume of the cone = π r² h/3
s = √ (5²+ h²)
Squaring on both sides, we will get,
s² = 15² = (5² + h²)
15² - 5² = h²
225 - 25 = 200 = h²
h = √200 = 14.1 ft
Volume = π × 5² × 14.1 / 3 = 117.5 π ft³
A distribution of measurements is relatively mound-shaped with a mean of 60 and a standard deviation of 14. Use this information to find the proportion of measurements in the given interval. between 46 and 74
Approximately 68.26% of the measurements fall between 46 and 74 in this distribution.
To find the proportion of measurements between 46 and 74 in a normal distribution with a mean (μ) of 60 and a standard deviation (σ) of 14, we can use the standard normal distribution (z-score) and the cumulative distribution function (CDF).
First, we need to convert the interval endpoints to z-scores using the formula:
z = (x - μ) / σ
Where x is the value in the interval, μ is the mean, and σ is the standard deviation.
For x = 46:
z₁ = (46 - 60) / 14
z₁ = -1
For x = 74:
z₂ = (74 - 60) / 14
z₂ = 1
Using the Excel functions:
=NORM.S.DIST(-1) and =NORM.S.DIST(1)
The probabilities are 0.1587 and 0.8413 respectively.
Now, we want the proportion of measurements between z₁ and z₂, which is:
Proportion = 0.8413 - 0.1587
≈ 0.6826
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Final answer:
Using the Empirical Rule for a normal distribution, approximately 68% of the measurements would fall between 46 and 74, as this range lies within one standard deviation above and below the mean of 60 in a distribution with a standard deviation of 14.
Explanation:
To find the proportion of measurements between 46 and 74 in a distribution with a mean of 60 and a standard deviation of 14, we can use the Empirical Rule, assuming the distribution is normal (bell-shaped). This rule states that approximately 68% of the data lies within one standard deviation of the mean, 95% within two, and more than 99% within three.
In this case, 46 is one standard deviation below the mean (60 - 14), and 74 is one standard deviation above the mean (60 + 14). So, we would expect approximately 68% of the measurements to lie between 46 and 74.
This is because the data is likely to be distributed symmetrically around the mean in a normal distribution, and the range given includes measurements falling within one standard deviation from the mean.
The age distribution of students at a community college is given below. Age (years) Number of students (f) Under 21 2890 21-24 2190 25-28 1276 29-32 651 33-36 274 37-40 117 Over 40 185 A student from the community college is selected at random. The events A and B are defined as follows. A = event the student is at most 32 B = event the student is at least 37 Are the events A and B disjoint? No Yes
Answer:
Are the events A and B disjoint? Yes
Step-by-step explanation:
Disjoint events are those events that cannot occur at the same time, i.e. for events X and Y to be disjoint, [tex]P(X\cap Y)=0[/tex].
The event A is defined as the number of students whose age is at most 32.
And event B is defined as the number of students whose age is at least 37.
The events A and B are disjoint events.
The sample space for event A consists of all the students of age group (under 21), (21 - 24), (25 - 28) and (29 - 32). Whereas the sample space for event B consists of all the students of age group (33 - 36), (37 - 40) and (Over 40).The sample space for the intersection of these two events is:
Sample space of (A ∩ B) = 0
As there are no common terms in both the sample.
Hence proved, events A and B are disjoint.
Find tea. Write your answer in simplest radical form
Answer:
2√6 ft
Step-by-step explanation:
Tan Ф = opposite/ adjacent
tan 60 = t / 2√2 ft
tan 60 = √3
t = (tan 60 )(2√2 ft)
t = (√3)(2√2 ft) = 2√6 ft
Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan.
Determine the sample space for the experiment.
Answer:
The sample space for the experiment is {(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}.
Step-by-step explanation:
Consider the provided information.
Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan.
The sample space for the experiment is:
{(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}
Hence, the possible outcomes are {(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}.
The circumference of a circle is 5picm.
What is the area of the circle?
A.) 6.25 pi cm2
B.) 2.5 pi cm2
C.) 25 pi cm2
D.) 10 pi cm2
We have N cars on a circular one-way road; they have the same make, same model, same year and the same fuel economy. The total amount of gas in all cars is sufficient to make the full circle.
Prove by induction that it is always possible to find a car that can make the full circle, taking gas from other cars as it passes them.
Answer: Satisfied for n=1, n=k and n=k+1
Step-by-step explanation:
The induction procedure involves two steps
First is
Basic Step
Here we consider that for the value n=1, there is one car and it will always make the full circle.
Induction Step
Since basic step is satisfied for n=1
Now we do it for n=k+1
Now according to the statement a car makes full circle by taking gas from other cars as it passes them. This means there are cars that are there to provide fuel to the car. So we have a car that can be eliminated i.e. it gives it fuels to other car to make full circle so it is always there.
Now ,go through the statement again that the original car gets past the other car and take the gas from it to eliminate it. So now cars remain k instead of k+1 as it's fuel has been taken. Now the car that has taken the fuel can make the full circle. The gas is enough to make a circle now.
So by induction we can find a car that satisfies k+1 induction so for k number of cars, we can also find a car that makes a full circle.
During the registration at the State University every semester, students in the college of business must have their courses approved by the college adviser. It takes the adviser an average of 2 minutes to approve each schedule, and students arrive at the adviser’s office at the rate of 28 per hour.17.How long does a student spend waiting on average for the adviser?A) 13 minutesB) 14 minutesC) 28 minutesD) 30 minutesE) none of the above
Answer:
Correct answer is option C i.e 28 minutes
Step-by-step explanation:
Number of students arriving at adviser's office per hour = x = 28
Number of students get approved = [tex]\frac{1}{2min}[/tex] = 30/hour
∴ y = 30
Number of students on average on waiting =Lq
Lq = [tex]\frac{x^{2} }{y(y-x)}[/tex]
= [tex]\frac{28^{2} }{30(30-28)}[/tex]
= 13.07
Average time student has to spend in
Waiting = Wq = [tex]\frac{x}{y(y-x)}[/tex]
= [tex]\frac{28}{30(30-28)}[/tex]
= 0.466 hours
= 28 minutes
Use the square roots property to solve the quadratic equation (y+150)2=50.
We can take the square root of both sides, adding a plus/minus sign of the right hand side:
[tex]\sqrt{(y+150)^2}=\pm\sqrt{50}\iff y+150 = \pm\sqrt{50}[/tex]
Then, we subtract 150 from both sides:
[tex]y=\pm\sqrt{50}-150[/tex]
So, the two solutions are
[tex]y_1 = \sqrt{50}-150,\quad y_2 = -\sqrt{50}-150[/tex]
A dreidel is a four-sided spinning top with the Hebrew letters nun, gimel, hei, and shin, one on each side. Each side is equally likely to come up in a single spin of the dreidel. Suppose you spin a dreidel three times. Calculate the probability of getting: (a) at least one nun? (b) exactly 2 nuns? (c) exactly 1 hei? (d) at most 2 gimels?
So, the probabilities are: (a) 37/64, (b) 3/64, (c) 27/64, (d) 57/64
In each spin, there are four possible outcomes (nun, gimel, hei, shin), and each outcome is equally likely.
(a) Probability of getting at least one nun:
The probability of getting no nuns in a single spin is 3/4. So, the probability of getting no nuns in three spins is [tex](3/4)^3[/tex]. Therefore, the probability of getting at least one nun is 1 - [tex](3/4)^3[/tex].
Probability of getting at least one nun:
1 - [tex](3/4)^3[/tex] = [tex]1-\frac{27}{64}=\frac{37}{64}[/tex] = 0.58
(b) Probability of getting exactly 2 nuns:
The probability of getting a nun in a single spin is 1/4. So, the probability of getting exactly 1 hei in three spins is [tex]\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2[/tex].
= [tex]3 \times \frac{1}{16} \times \frac{3}{4}=\frac{3}{64}[/tex] = 0.05
(c) Probability of getting exactly 1 hei:
The probability of getting a hei in a single spin is 1/4. So, the probability of getting exactly 1 hei in three spins is [tex]\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2[/tex]
= [tex]3 \times \frac{1}{4} \times \frac{9}{16}=\frac{27}{64}[/tex] = 0.42
(d) Probability of getting at most 2 gimels:
The probability of getting 0 gimels is [tex](3/4)^3[/tex]. The probability of getting 1 gimel is [tex]\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2[/tex]. The probability of getting 2 gimels is [tex]\left(\begin{array}{l}\frac{3}{2} \\\end{array}\right)\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)[/tex].
Add these probabilities to get the total probability.
[tex]\left(\frac{3}{4}\right)^3+\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2+\left(\begin{array}{l}\frac{3}{2} \\\end{array}\right)\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)[/tex]
[tex]=\frac{27}{64}+\frac{27}{64}+\frac{3}{64}=\frac{57}{64}[/tex] = 0.9
Calculating probabilities of specific outcomes when spinning a dreidel multiple times.
Dreidel Probability Calculations:
(a) Probability of getting at least one nun: 1 - Probability of getting no nuns = 1 - [tex](3/4)^3[/tex].(b) Probability of getting exactly 2 nuns: Combination of outcomes with exactly 2 nuns / Total possible outcomes = (3 choose 2) x [tex](1/4)^2[/tex] x (3/4).(c) Probability of getting exactly 1 hei: Combination of outcomes with exactly 1 hei / Total possible outcomes = 3 x (1/4) x [tex](3/4)^2[/tex].(d) Probability of getting at most 2 gimels: Sum of probabilities of getting 0, 1, or 2 gimels.The mean waiting time at the drive-through of a fast-food restaurant from the time the food is ordered to when it is received is 85 seconds. A manager devises a new system that he believes will decrease the wait time. He implements the new system and measures the wait time for 10 randomly sampled orders. They are provided below:
109 67 58 76 65 80 96 86 71 72
Assume the population is normally distributed.
(a) Calculate the mean and standard deviation of the wait times for the 10 orders.
(b) Construct a 99% confidence interval for the mean waiting time of the new system.
Answer:
a) And if we replace we got: [tex]\bar X= 78[/tex]
[tex] s = 15.391[/tex]
b) [tex]78-3.25\frac{15.391}{\sqrt{10}}=62.182[/tex]
[tex]78-3.25\frac{15.391}{\sqrt{10}}=93.818[/tex]
So on this case the 99% confidence interval would be given by (62.182;93.818)
Step-by-step explanation:
Dataset given: 109 67 58 76 65 80 96 86 71 72
Part a
For this case we can calculate the sample mean with the following formula:
[tex]\bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]
And if we replace we got: [tex]\bar X= 78[/tex]
And the deviation is given by:
[tex] s =\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}[/tex]
And if we replace we got [tex] s = 15.391[/tex]
Part b
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:
[tex]df=n-1=10-1=9[/tex]
Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,9)".And we see that [tex]t_{\alpha/2}=3.25[/tex]
Now we have everything in order to replace into formula (1):
[tex]78-3.25\frac{15.391}{\sqrt{10}}=62.182[/tex]
[tex]78-3.25\frac{15.391}{\sqrt{10}}=93.818[/tex]
So on this case the 99% confidence interval would be given by (62.182;93.818)
In response to a survey question about the number of hours daily spent watching TV, the responses by the eight subjects who identified themselves as Hindu were 2 , 2 , 1 , 3 , 1 , 0 , 4 , 1
a. Find a point estimate of the population mean for Hindus.
--------------(Round to two decimal places as needed)
b. The margin of error at the 95% confidence level for this point estimate is 0.89. Explain what this represents.
The margin of error indicates we can be__%confident that the sample mean falls within __ of the _____(population mean/ standard error/ sample mean)
Answer:
a) [tex] \bar X = \frac{2+2+1+3+1+0+4+1}{8}= 1.75[/tex]
b) The margin of error indicates we can be 95%confident that the sample mean falls within 0.89 of the population mean
Step-by-step explanation:
Part a
The best point of estimate for the population mean is the sample mean given by:
[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]
Since is an unbiased estimator [tex] E(\bar X) = \mu[/tex]
Data given: 2 , 2 , 1 , 3 , 1 , 0 , 4 , 1
So for this case the sample mean would be:
[tex] \bar X = \frac{2+2+1+3+1+0+4+1}{8}= 1.75[/tex]
Part b
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
[tex]\bar X[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s represent the sample standard deviation
n represent the sample size
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
The margin of error is given by this formula:
[tex] ME=t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (2)
And for this case we know that ME =0.89 with a confidence of 95%
So then the limits for our confidence level are:
[tex] Lower= \bar X -ME= 1.75- 0.89=0.86[/tex]
[tex] Upperr= \bar X +ME= 1.75+0.89=2.64[/tex]
So then the best answer for this case would be:
The margin of error indicates we can be 95%confident that the sample mean falls within 0.89 of the population mean
Suppose that out of 20% of all packages from Amazon are delivered by UPS, 12% of the packages that are delivered by UPS weighs 2 lbs or more. Also, 8% of the packages that are not delivered by UPS weighs less than 2 lbs.
a. What is the probability that a package is delivered by UPS if it weighs 2 lbs or more?
b. What is the probability that a package is not delivered by UPS if it weighs 2 lbs or more?
Answer:
(a) Probability that a package is delivered by UPS if it weighs 2 lbs or more = 0.0316.
(b) Probability that a package is not delivered by UPS if it weighs 2 lbs or more = 0.9684 .
Step-by-step explanation:
We are given that 20% of all packages from Amazon are delivered by UPS, from which 12% of the packages that are delivered by UPS weighs 2 lbs or more and 8% of the packages that are not delivered by UPS weighs less than 2 lbs.
Firstly Let A = Package from Amazon is delivered by UPS.
B = Packages that are delivered by UPS weighs 2 lbs or more.
So, P(A) = 0.2 and P(A') = {Probability that package is not delivered by UPS}
P(A') = 1 - 0.2 = 0.8
P(B/A) = 0.12 {means Probability that package weight 2 lbs or more given it
is delivered by UPS}
P(B'/A') = 0.08 [means Probability that package weight less than 2 lbs given
it is not delivered by UPS}
Since, P(B/A) = [tex]\frac{P(A\bigcap B)}{P(A)}[/tex] , [tex]P(A\bigcap B)[/tex] = P(B/A) * P(A) = 0.12 * 0.2 = 0.024 .
Also P(B) { Probability that package weight 2 lbs or more} is given by;
Probability that package weight 2 lbs or more and it delivered by UPS.Probability that package weight 2 lbs or more and is not delivered by UPS.So, P(B) = [tex]P(B\bigcap A) + P(B\bigcap A')[/tex] = P(B/A) * P(A) + P(B/A') * P(A')
= 0.12 * 0.2 + 0.92 * 0.8 { Here P(B/A') = 1 - P(B'/A') = 1 - 0.08 = 0.92}
= 0.76
(a) Probability that a package is delivered by UPS if it weighs 2 lbs or more is given by P(A/B);
P(A/B) = [tex]\frac{P(A\bigcap B)}{P(B)}[/tex] = [tex]\frac{0.024}{0.76}[/tex] = 0.0316
(b) Probability that a package is not delivered by UPS if it weighs 2 lbs or more = 1 - P(A/B) = 1 - 0.0316 = 0.9684 .