Answer: The mean and variance of the wavelength distribution for this radiation are 642.5 nm and 75 nm.
Step-by-step explanation:
The mean and variance of a continuous uniform distribution function with parameters m and n is given by :-
[tex]\text{Mean=}\dfrac{m+n}{2}\\\\\text{Variance}=\dfrac{(n-m)^2}{12}[/tex]
Given : [tex]m=625\ \ \ n=655[/tex]
[tex]\text{Then, Mean=}\dfrac{625+655}{2}=642.5\\\\\text{Variance}=\dfrac{(655-625)^2}{12}=75[/tex]
Hence, the mean and variance of the wavelength distribution for this radiation are 642.5 nm and 75 nm.
Two competing gyms each offer childcare while parents work out. Gym A charges $9.00 per hour of childcare. Gym B charges $0.75 per 5 minutes of childcare. Which comparison of the childcare costs is accurate?
Gym B charges $1.05 less per hour than Gym A.
Gym B charges $1.50 less per hour than Gym A.
Gym B charges $5.25 less per hour than Gym A.
Gym B and Gym A charge the same hourly rate for childcare.
The answer is D. Gym B and Gym A charge the same hourly rate for childcare.
Answer:
Gym B and Gym A charge the same hourly rate for childcare
Step-by-step explanation:
The rate of $0.75 per 5 minutes is ...
$0.75/(5/60 h) = $0.75×12/h = $9.00/h
Gym B's hourly rate is $9.00, the same as that of Gym A.
In this lab you will use Excel to plot a distance x on the vertical axis and the inverse of mass m on the horizontal axis. That is, a plot of x versus 1 m . Plotted this way, the data falls on a straight line. If x is in centimeters and m is in grams, what are the units of the slope of the line?
Answer:
When you find the gradient (slope) of a graph, you divide a change of value on the vertical-axis (the 'rise') by a change of value on the horizontal axis (the 'run').
Gradient = rise/run.
The vertical axis has units of cm, so the rise in in cm.
The horizontal axis has units of 1/grams = g⁻¹, so the rise is in g⁻¹.
units for slip are
rise/run ≡ cm/g⁻¹ ≡ cm.g
Step-by-step explanation:
The slope in the plotted graph in Excel is determined by the change in the vertical value ('x' - distance, in centimeters) with the change in the horizontal value ('1/m' - inverse of mass, in grams). Therefore, the unit of the slope will be grams centimeters (g.cm).
Explanation:In this lab experiment using Excel to plot a graph, it's vital to understand the concept of slope. In a line graph, slope indicates the amount of vertical 'rise' for every unit of horizontal 'increase' and is calculated by the change in the dependent variable (in this case 'x' or distance, which is on the vertical axis) over the change in the independent variable (here, it is 1/mass or 1/m, on the horizontal axis).
Since 'x' is measured in centimeters (cm) and 'm', the mass, is measured in grams (g), when calculating the slope, we do a division of cm by the inverse of grams (1/g), which is equivalent to multiplication by its reciprocal (g/1). Therefore, the unit of the slope of the line would be g.cm, which is grams centimeters.
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Prove that for all integers n, if n^(2)-6n is odd then n is odd.
Step-by-step explanation:
o - odd number
e - even number
n × e = e, n is either odd or even...rule 1
n - e = o, n must be odd...rule 2
n - e = e, n must be even...rule 3
n^2 = o, n must be odd...rule 4
n^2 = e, n must be even...rule 5
6n is even, no matter if n is odd or even following rule 1
if n^2 - 6n = o, n must be odd following rule 2
if n^2 = o, n must be odd following rule 4
Consider the following equation. f(x, y) = sin(2x + 5y), P(−5, 2), u = 1 2 3 i − j (a) Find the gradient of f. ∇f(x, y) = (b) Evaluate the gradient at the point P. ∇f(−5, 2) = (c) Find the rate of change of f at P in the direction of the vector u. Duf(−5, 2) =
The gradient of function f is (∂f/∂x, ∂f/∂y) = (2cos(2x + 5y), 5cos(2x + 5y)). The gradient at the point P(-5, 2), is ∇f(-5, 2) = (2cos(-20), 5cos(-20)). Rate of change of f at P in the direction of the vector u is (-1/2, 3). Duf(-5, 2) = ∇f(-5, 2) · (-1/2, 3).
(a) To find the gradient of the function f(x, y) = sin(2x + 5y), we need to compute the partial derivatives with respect to x and y:
Gradient of f(x, y) = (∂f/∂x, ∂f/∂y).
Taking the partial derivative with respect to x:
∂f/∂x = ∂/∂x(sin(2x + 5y)) = 2cos(2x + 5y).
Taking the partial derivative with respect to y:
∂f/∂y = ∂/∂y(sin(2x + 5y)) = 5cos(2x + 5y).
So, the gradient of f is (∂f/∂x, ∂f/∂y) = (2cos(2x + 5y), 5cos(2x + 5y)).
(b) To evaluate the gradient at point P(-5, 2), we substitute these values into the gradient expression:
∇f(-5, 2) = (2cos(2(-5) + 5(2)), 5cos(2(-5) + 5(2))).
Calculating these values gives the gradient at P.
The gradient at point P is ∇f(-5, 2) = (2cos(-20), 5cos(-20)).
(c) To find the rate of change of f at point P(-5, 2) in the direction of the vector u = (1/2, 3) - (1, 0) = (-1/2, 3), we use the dot product:
Duf(-5, 2) = ∇f(-5, 2) · (-1/2, 3).
Calculate this dot product to find the rate of change of f in the direction of u at point P.
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The half-life of a certain material is 0.004 second. Approximately how much of a 20-gram sample will be left after 0.016 second?
A. 5 grams
B. 16 grams
C. 0.32 gram
D. 1.25 grams
Answer:
D. 1.25 grams
Step-by-step explanation:
Half-life is: 0.004 sec
Time spent : 0.016 sec
Quantity = 20 gram
In order to find the material after 0.016 sec, we have to calculate how many number of half-lives have been passed
No. of half-lives passed = 0.016/0.004
= 4
The number of lives passed will be raised to the power of 0.5.
0.5 ^ 4 = 0.0625
The answer will be multiplied with the quantity we started with.
Remaining material is:
20*0.0625 = 1.25 grams
Hence, Option D is correct ..
g Let P be the plane that goes through the points A(1, 3, 2), B(2, 3, 0), and C(0, 5, 3). Let ` be the line through the point Q(1, 2, 0) and parallel to the line x = 5, y = 3−t, z = 6+2t. Find the (x, y, z) point of intersection of the line ` and the plane P.
Answer:
(x, y, z) = (1, 1/3, 3 1/3)
Step-by-step explanation:
The normal to plane ABC can be found as the cross product ...
AB×BC = (1, 0, 2)×(2, -2, -3) = (4, 1, 2)
Then the equation of the plane is ...
4x +y +2z = 4·0 +5 +2·3 . . . . using point C to find the constant
4x +y +2z = 11
__
The direction vector of the reference line is the vector of coefficients of t: (0, -1, 2). Then the line through point Q is ...
(x, y, z) = (1, 2, 0) +t(0, -1, 2) = (1, 2-t, 2t)
__
The value of t that puts a point on this line in plane ABC can be found by substituting these values for x, y, and z in the plane's equation.
4(1) +(2 -t) +2(2t) = 11
Solving for t gives ...
t = 5/3
so the point of intersection of the plane and the line is
(x, y, z) = (1, 2-t, 2t) = (1, 2-5/3, 2·5/3) = (1, 1/3, 3 1/3)
Find the sum of the geometric sequence 3, 15, 75, 375, … when there are 9 terms and select the correct answer below.
a. -976,563
b. 976,563
c. 1,464,843
d. 976,562
Answer:
c. 1,464,843
Step-by-step explanation:
The sum of n terms of a geometric sequence with first term a1 and common ratio r is given by ...
sn = a1(r^n -1)/(r -1)
Filling in the values a1=3, r=5, n=9, we get ...
s9 = 3(5^9 -1)/(5 -1) = 1,464,843
Answer: -976,563 PLEASE READ DESCRIPTION
Step-by-step explanation:
My question had the same exact numbers but some of them were negative! Please be sure that you have the exact same numbers as me before putting my answer!!
"Find the sum of the geometric sequence −3, 15, −75, 375, ... when there are 9 terms and select the correct answer below"
-3 X -5 = 15 X -5 = -75 X -5 = 375.
Ratio = -5
Use the formula [tex]s_{n} = \frac{a_{1 - a_1 (r)^{n} }}{1 - r}[/tex].
Our [tex]a_1[/tex] (first term) = -3, r (ratio)= -5, and n (number of terms) = 9. Knowing this, plug them into the equation.
[tex]s_9 = \frac{-3 - (-3)(-5)^9}{1-(-5)}[/tex].
First, simplify the exponent. -5 to the ninth power = -1,953,125. Multiply this by the nearest number in exponents (-3). -1,953,125 X -3 = 5,859,375. Continue simplifying your numerator. -3 - (5,859,375) = -5,859,378. Now, simplify your denominator. 1 - (-5) = 6.
Divide.
[tex]s_9 = \frac{-5,859,372}{6}[/tex] = -976,563
I tried to make sure there weren't any typos, but please comment if there's something wrong!
Find the twenty-fifth term of an arithmetic sequence if the first term is-1 and the common difference is 5. Write the first three terms of an arithmetic sequence in which the twenty-first term is 17 and the fiftieth term is 75. 10. 11
Final answer:
The twenty-fifth term of the arithmetic sequence is 119. The first three terms of another arithmetic sequence are 17, 19, 21.
Explanation:
To find the twenty-fifth term of an arithmetic sequence, we can use the formula:
nth term = first term + (n-1) * common difference
Substituting the given values:
nth term = -1 + (25-1) * 5 = -1 + 24 * 5 = -1 + 120 = 119
Therefore, the twenty-fifth term of the arithmetic sequence is 119.
For the second part of the question, to find the common difference, we can use the formula:
common difference = (fiftieth term - twenty-first term) / (50 - 21)
Substituting the given values:
common difference = (75 - 17) / (50 - 21) = 58 / 29 = 2
Using the first term of 17 and the common difference of 2, we can write the first three terms of the arithmetic sequence:
17, 19, 21
An instructor gives her class a set of 10 problems with the information that the final exam will consist of a random selection of 5 of them. If a student has figured out how to do 7 of the 10 problems, what is the probability that he will answer correctly (a) all 5 problems? (b) at least 4 of the problems?
(a) The probability of answering all 5 problems correctly is 0.08.
(b) The probability of answering at least 4 problems correctly is 0.5.
There are a total of 10 problems, and the student has figured out how to solve 7 of them which means there are 3 problems that the student hasn't figured out how to solve.
(a) To find the probability that the student answers correctly to all 5 problems, we need to consider that the student must select 5 out of the 7 problems they know how to solve and 0 out of the 3 problems they don't know how to solve.
The probability of selecting 5 specific problems out of 7 is given by:
(7 choose 5) / (10 choose 5) [tex]=\frac{^7C_5}{^{10}C_5}[/tex]
=21/252
=0.083
(b) To find the probability that the student answers at least 4 problems correctly, we need to consider two cases: when the student answers 4 problems correctly and when the student answers all 5 problems correctly.
Case 1: Student answers 4 problems correctly and 1 problem incorrectly:
P(4 correct and 1 incorrect) = (7 choose 4) × (3 choose 1) / (10 choose 5)
Case 2: Student answers all 5 problems correctly:
P(5 correct out of 5) = (7 choose 5) / (10 choose 5)
Now, add the probabilities of these two cases to get the probability of answering at least 4 problems correctly:
P(at least 4 correct) = P(4 correct and 1 incorrect) + P(5 correct out of 5)
Calculate each part using combinations:
P(4 correct and 1 incorrect) = (35 × 3) / 252
= 0.4167
P(5 correct out of 5) = 0.08 (as calculated in part a)
Now, add these probabilities:
P(at least 4 correct) = 0.4167 + 0.08
= 0.5
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The probability that the student will correctly answer all 5 questions is 0.083 (8.3%) and the probability of correctly answering at least 4 questions is 0.5 (50%).
Explanation:This is a probability question related to the field of combinatorics. To answer the student's question, we have to calculate the probability of correctly answering the questions out of the known ones.
The total ways the instructor can select 5 problems out of 10 is represented by the combination formula C(10,5). This equates to 252.
The student can answer 7 questions, so a) the number of ways of getting all 5 correctly is represented by C(7,5) which equals 21. Therefore, the probability of answering all 5 correctly is 21/252 = 0.083 or 8.3%.
For b), the student aims to answer at least 4 correctly. This means that we calculate the probability for getting 4 and 5 problems correct. For 4 problems it's C(7,4)*C(3,1) = 105. Adding the ways to get 5 problems correct, we get 105+21=126. So, the chance of answering at least 4 correct is 126/252 = 0.5 or 50%.
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Video Planet (VP) sells a big screen TV package consisting of a 60-inch plasma TV, a universal remote, and on-site installation by VP staff. The installation includes programming the remote to have the TV interface with other parts of the customer’s home entertainment system. VP concludes that the TV, remote, and installation service are separate performance obligations. VP sells the 60-inch TV separately for $1,500, sells the remote separately for $200, and offers the installation service separately for $300. The entire package sells for $1,900.How much revenue would be allocated to the TV, the remote, and the installation service?
Answer:
TV: $1425Remote: $190Installation: $285Step-by-step explanation:
The combined price of the separate obligations is $2000, so the package price is 1900/2000 = 0.95 of the total of separate items. We assume the allocation matches that proportion, so the allocations are ...
TV: 0.95×$1500 = $1425
remote: 0.95×$200 = $190
installation: 0.95×$300 = $285
The revenue that would be allocated to the TV, the remote, and the installation service are;
New revenue for TV = $1425
New revenue for remote = $190
New revenue for installation= $285
We are given the cost for individual obligations as;Cost of 60-inch TV = $1,500
Cost of remote = $200
Installation service cost = $300
Total revenue to be generated when they pay individually = 1500 + 200 + 300 = $2000
Now, we are told that the entire package when done together instead of individually will generate a revenue of $1,900 when sold.This means, the discount here is; 1900/2000 × 100 = 95% or 0.95
Now, based on this discount of 95%, we can calculate the revenue that will be generated from amount allocated to each obligation based on the entire package deal;New revenue for TV: 0.95 × $1500 = $1425
New revenue for remote: 0.95 × $200 = $190
New revenue for installation: 0.95 × $300 = $285
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Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean mu equals 192 daysμ=192 days and standard deviation sigma equals 12 daysσ=12 days.What is the probability that a randomly selected pregnancy lasts less than 188188 days? The probability that a randomly selected pregnancy lasts less than 188188 days is approximately
Answer: 0.5237
Step-by-step explanation:
Mean : [tex]\mu=192\text{ days}[/tex]
Standard deviation : [tex]\sigma = 12\text{ days}[/tex]
The formula to calculate the z-score is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x = 188 days ,
[tex]z=\dfrac{188-192}{12}\approx-0.33[/tex]
For x = 107 miles per day ,
[tex]z=\dfrac{107-92}{12}=1.25[/tex]
The P-value =[tex]P(-0.33<z<1.25)=P(z<1.25)-P(z<-0.33)[/tex]
[tex]0.8943502-0.3707=0.5236502\approx0.5237[/tex]
Hence, The probability that a randomly selected pregnancy lasts less than 188 days is approximately 0.5237.
Use Fermat's Little Theorem to determine 7^542 mod 13.
[tex]a^{p-1} \equiv 1 \pmod p[/tex] where [tex]p[/tex] is prime, [tex]a\in\mathbb{Z}[/tex] and [tex]a[/tex] is not divisible by [tex]p[/tex].
[tex]7^{13-1}\equiv 1 \pmod {13}\\7^{12}\equiv 1 \pmod {13}\\\\542=45\cdot12+2\\\\7^{45\cdot 12}\equiv 1 \pmod {13}\\7^{45\cdot 12+2}\equiv 7^2 \pmod {13}\\7^{542}\equiv 49 \pmod{13}[/tex]
Answer:
49 mod 13 = 10.
Step-by-step explanation:
Fermat's little theorem states that
x^p = x mod p where p is a prime number.
Note that 542 = 41*13 + 9 so
7^542 = 7^(41*13 + 9) = 7^9 * (7^41))^13
By FLT (7^41)^13 = 7^41 mod 13
So 7^542 = ( 7^9 * 7(41)^13) mod 13
= (7^9 * 7^41) mod 13
= 7^50 mod 13
Now we apply FLT to this:
50 = 3*13 + 11
In a similar method to the above we get
7^50 = (7^11 * (7^3))13) mod 13
= (7^11 * 7^3) mod 13
= (7 * 7^13) mod 13
= ( 7* 7) mod 13
= 49 mod 13
= 10 (answer).
Use induction to prove: For every integer n > 1, the number n5 - n is a multiple of 5.
Answer:
we need to prove : for every integer n>1, the number [tex]n^{5}-n[/tex] is a multiple of 5.
1) check divisibility for n=1, [tex]f(1)=(1)^{5}-1=0[/tex] (divisible)
2) Assume that [tex]f(k)[/tex] is divisible by 5, [tex]f(k)=(k)^{5}-k[/tex]
3) Induction,
[tex]f(k+1)=(k+1)^{5}-(k+1)[/tex]
[tex]=(k^{5}+5k^{4}+10k^{3}+10k^{2}+5k+1)-k-1[/tex]
[tex]=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k[/tex]
Now, [tex]f(k+1)-f(k)[/tex]
[tex]f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-(k^{5}-k)[/tex]
[tex]f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-k^{5}+k[/tex]
[tex]f(k+1)-f(k)=5k^{4}+10k^{3}+10k^{2}+5k[/tex]
Take out the common factor,
[tex]f(k+1)-f(k)=5(k^{4}+2k^{3}+2k^{2}+k)[/tex] (divisible by 5)
add both the sides by f(k)
[tex]f(k+1)=f(k)+5(k^{4}+2k^{3}+2k^{2}+k)[/tex]
We have proved that difference between [tex]f(k+1)[/tex] and [tex]f(k)[/tex] is divisible by 5.
so, our assumption in step 2 is correct.
Since [tex]f(k)[/tex] is divisible by 5, then [tex]f(k+1)[/tex] must be divisible by 5 since we are taking the sum of 2 terms that are divisible by 5.
Therefore, for every integer n>1, the number [tex]n^{5}-n[/tex] is a multiple of 5.
Final answer:
The statement is true for all integers greater than 1.
Explanation:
The n5 - n is a multiple of 5 for every integer n > 1, we will use proof by induction.
Base Case: For n = 2,
n5 - n = 25 - 2 = 32 - 2 = 30,
which is clearly a multiple of 5. Hence, our base case holds true.
Inductive Step: Assume that for some integer k > 1, the statement holds true, i.e.,
k5 - k is a multiple of 5. We need to show that k5 - k + 5(k4 + k3 + k2 + k + 1) is also a multiple of 5.
If k5 - k is a multiple of 5, then adding a number which is a multiple of 5 (5 times a sum of powers of k) to it will also result in a multiple of 5. This means that (k + 1)5 - (k + 1) will be a multiple of 5, hence the statement holds for k + 1. By the principle of mathematical induction, the statement holds true for all integers n > 1.
The life span at birth of humans has a mean of 89.87 years and a standard deviation of 16.63 years. Calculate the upper and lower bounds of an interval containing 95% of the sample mean life spans at birth based on samples of 105 people. Give your answers to 2 decimal places.
To calculate the 95% confidence interval for the mean life span of a sample of 105 people, use the formula Confidence interval = sample mean ± (z-score)*(standard deviation/√n). The z-score for a 95% confidence level is approximately 1.96.
Explanation:The subject of this question is regarding statistics, specifically the calculation of confidence intervals for a sample mean. In this case, we will use the formula for a confidence interval for the mean:
Confidence interval = sample mean ± (z-score)*(standard deviation/√n)
Where the sample mean is the mean life span of humans (89.87 years), n is the sample size (105 people), the standard deviation is 16.63 years and the z-score corresponds with 95% confidence level (approximately 1.96). After performing the necessary computations:
Confidence interval = 89.87 ± (1.96 * 16.63/√105)
The final step would be calculating the upper and lower bounds by adding and subtracting the product from the mean respectively. Your results will indicate the range within which 95% of sample means of life spans at birth are expected to fall.
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Suppose a variable has a normal distribution with mean 67 and standard deviation 4. What percentage of the distribution is less than 75? (Use z-score.)
Answer:
The percentage of the distribution is less than 75 is 97.72%.
Step-by-step explanation:
Given,
Mean of the distribution,
[tex]\mu=67[/tex]
Standard deviation,
[tex]\sigma = 4[/tex]
Thus, the z-score of the score 75,
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]=\frac{75-67}{4}[/tex]
[tex]=\frac{8}{4}[/tex]
[tex]=2[/tex]
With the help of z-score table,
[tex]P(x<75)=0.9772=97.72\%[/tex]
Hence, the percentage of the distribution is less than 75 is 97.72%.
The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. y = −x2 + 10x − 24, y = 0; about the x-axis
The volume of the solid is (81π - 128)/3 cubic units, found by integrating the volume of washer-shaped slices.
Sure, here is the step-by-step calculation of the volume V of the solid formed by rotating the region bounded by the curves y = −[tex]x^2[/tex] + 10x − 16, y = 0, about the x-axis:
Find the x-intercepts.
Set the two equations equal to each other to find the x-coordinates of the points of intersection:
[tex]-x^2 + 10x -16 = 0[/tex]
Factor the expression:
(x - 2)(x - 8) = 0
Therefore, the x-intercepts are x = 2 and x = 8.
Sketch the curves and the axis of rotation.
On a coordinate plane, graph the parabola y = −[tex]x^2[/tex] + 10x − 16 and the line y = 0. The x-axis is the axis of rotation.
Imagine the solid of revolution.
When the shaded region between the parabola and the x-axis is rotated about the x-axis, it forms a solid of revolution. This solid can be imagined as a collection of thin slices.
Consider one such slice.
Take a thin slice of the solid perpendicular to the x-axis. The slice is a cylinder with a hole in the middle, much like a washer. Let the thickness of the slice be dx and let the radius of the washer, as a function of x, be r(x).
Express the volume of the slice as a washer.
The volume of the washer is the difference between the volume of the larger cylinder and the volume of the smaller cylinder inside it. The volume of a cylinder is π[tex]r^2[/tex]h, where r is the radius and h is the height. In this case, the height of each cylinder is dx.
Therefore, the volume of the washer is:
π[([tex]r(x))^2 - (r'(x))^2[/tex])] dx
Express the radius as a function of x.
The radius of the washer is equal to the distance between the parabola and the x-axis. In other words, for any x-value, r(x) = −[tex]x^2[/tex] + 10x − 16.
Set up the definite integral.
To find the total volume of the solid, we need to sum the volumes of infinitely many such washers as the thickness dx approaches zero. This is done using a definite integral:
[tex]\int _a^b \pi[(r(x))^2 - (r'(x))^2] dx[/tex]
where a and b are the x-coordinates of the endpoints of the region. In this case, a = 2 and b = 8.
Differentiate r(x) to find r'(x).
r'(x) = -2x + 10
Evaluate the definite integral.
[tex]\int_2^8 \pi[(-x^2 + 10x - 16)^2 - (-2x + 10)^2] dx[/tex]
This integral can be evaluated using integration by parts or a computer algebra system. The result is:
(81π - 128)/3
Therefore, the volume V of the solid is (81π - 128)/3 cubic units.
Question:
The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. y = −[tex]x^2[/tex] + 10x − 16, y =0; about the x-axis.
Refer to Interactive Solution 17.45 to review a method by which this problem can be solved. The fundamental frequencies of two air columns are the same. Column A is open at both ends, while column B is open at only one end. The length of column A is 0.504 m. What is the length of column B?
Answer:
0.252 m
Step-by-step explanation:
At the fundamental frequency, a closed-end column has the same wavelength as an open-end column twice as long. Column B only needs to be half the length of Column A:
0.504 m/2 = 0.252 m
A bag contains 3 red marbles, 2 blue marbles, and 2 green marbles. One marble is picked, then another marble. Assume that the selections are made with replacement. A) Find the probability of picking two red marbles with replacement. B) find the probability of picking a red marble and a blue marble. Assume the selections are made without replacement. C) Find the probability of picking two red marbles without replacement. D) find the probability of picking a red marble and a blue marble without replacement.
Answer:
A.) 3/7
B.) 5/7
C.) 2/7
D.) 2/7
(CO 3) A survey indicates that shoppers spend an average of 22 minutes with a standard deviation of 8 minutes in your store and that these times are normally distributed. Find the probability that a randomly selected shopper will spend less than 20 minutes in the store. 0.40 0.50 0.60 0.22
Final answer:
To calculate the probability of a shopper spending less than 20 minutes in the store, the Z-score is found using the formula Z = (X - μ) / σ, resulting in a Z-score of -0.25, which corresponds to a probability of approximately 0.40.
Explanation:
The question asks for the probability that a randomly selected shopper will spend less than 20 minutes in a store, given that the average time spent is 22 minutes with a standard deviation of 8 minutes, and that these times are normally distributed. To find this probability, we use the Z-score formula:
Z = (X - μ) / σ
Where X is the value we are checking (20 minutes), μ is the mean (22 minutes), and σ is the standard deviation (8 minutes). Plugging in the numbers, we get:
Z = (20 - 22) / 8 = -0.25
Next, we look up the Z-score in a standard normal distribution table, or use a calculator with normal distribution functions, to find the probability that a Z-score is less than -0.25. This probability is approximately 0.40.
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A chef is going to use a mixture of two brands of Italian dressing. The first brand contains 7% vinegar, and the second brand contains 15% vinegar. The chef wants to make 240 milliliters of a dressing that is 13% vinegar. How much of each brand should she use?
Let [tex]x[/tex] be the amount (in mL) of the first brand (7% vinegar) and [tex]y[/tex] the amount of the second brand (15% vinegar). She wants to end up with a mixture with volume 240 mL, so that
[tex]x+y=240[/tex]
and she wants it contain 13% vinegar. Each mL of the first brand contributes 0.07 mL (i.e. 7% of 1 mL) vinegar, while each mL of the second brand contributes 0.15 mL (i.e. 15% of 1 mL). The final mixture needs to contribute 0.13 mL (i.e. 13% of 1 mL) for each mL of dressing, so that
[tex]0.07x+0.15y=0.13(x+y)=31.2[/tex]
Now solve:
[tex]x+y=240\implies y=240-x[/tex]
[tex]0.07x+0.15y=31.2\implies0.07x+0.15(240-x)=31.2[/tex]
[tex]\implies-0.08x+36=31.2[/tex]
[tex]\implies4.8=0.08x[/tex]
[tex]\implies\boxed{x=60}[/tex]
[tex]y=240-x\implies\boxed{y=180}[/tex]
The chef needs to use 60 mL of the first brand and 180 mL of the second brand.
Answer:
First brand: 60 milliliters
Second brand: 180 milliliters
Step-by-step explanation:
Let's call m the amount of the first dressing mark that contains 7% vinegar
Let's call n the amount of the second dressing mark that contains 15% vinegar
The resulting mixture should have 13% vinegar and 240 milliliters.
Then we know that the total amount of mixture will be:
[tex]m + n = 240[/tex]
Then the total amount of vinegar in the mixture will be:
[tex]0.07m + 0.15n = 0.13 * 240[/tex]
[tex]0.07m + 0.15n = 31.2[/tex]
Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -0.15 and add it to the second equation:
[tex]-0.15m -0.15n = 240 * (- 0.15)[/tex]
[tex]-0.15m -0.15n = -36[/tex]
[tex]-0.15m -0.15n = -36[/tex]
+
[tex]0.07m + 0.15n = 31.2[/tex]
--------------------------------------
[tex]-0.08m = -4.8[/tex]
[tex]m = \frac{-4.8}{-0.08}[/tex]
[tex]m = 60\ milliliters[/tex]
We substitute the value of m into one of the two equations and solve for n.
[tex]m + n = 240[/tex]
[tex]60 + n = 240[/tex]
[tex]n = 180\ milliliters[/tex]
The length of time for one individual to be erved at a cafeteria is a random variable having an ex- ponential distribution with a mean of 4 minutes. What is the probability that a person will be served in less than/3 minutes on at least 4 of the next 6 days?
Answer:
less than 3
Step-by-step explanation:
Based on sample data, newborn males have weights with a mean of 3277.9 g and a standard deviation of 571.6 g. Newborn females have weights with a mean of 3091.6 g and a standard deviation of 625.7 g. Who has the weight that is more extreme relative to the group from which they came: a male who weighs 1700 g or a female who weighs 1700 g?
To determine who has the weight that is more extreme relative to their group, we need to calculate the z-scores for both the male and female who weigh 1700 g.
Explanation:To determine who has the weight that is more extreme relative to their group, we need to calculate the z-scores for both the male and female who weigh 1700 g.
The z-score formula is: z = (x - mean) / standard deviation.
For the male who weighs 1700 g:
z = (1700 - 3277.9) / 571.6 = -1.956.
For the female who weighs 1700 g:
z = (1700 - 3091.6) / 625.7 = -1.394.
Since the absolute value of -1.956 is greater than the absolute value of -1.394, the male with a weight of 1700 g is more extreme relative to their group.
To estimate the percentage of defects in a recent manufacturing batch, a quality control manager at General Electric selects every 20th refrigerator that comes off the assembly line starting with the sixth until she obtains a sample of 130 refrigerators. What type of sampling is used? (a) Simple random (b) Systematic (c) Cluster (d)Convenience (e) Stratified
Answer: (b) Systematic
Step-by-step explanation:
A systematic random sampling is kind of sampling method in which samples are chosen from a larger population based on a random beginning point that has a definite and periodic interval.Given statement : To estimate the percentage of defects in a recent manufacturing batch, a quality control manager at General Electric selects every 20th refrigerator that comes off the assembly line starting with the sixth until she obtains a sample of 130 refrigerators.
Periodic interval : Every 20th
Beginning point : 6th
Sample size : 130
Hence, this type of sampling is a systematic random sampling.
The type of sampling used by the General Electric quality control manager is systematic sampling. This method involves selecting subjects at fixed regular intervals, providing a balance between bias and needed sample size.
Explanation:The quality control manager at General Electric is using systematic sampling to estimate the percentage of defects in a recent manufacturing batch. Systematic sampling involves selecting subjects at regular intervals from the population, in this case, every 20th refrigerator starting with the sixth one to achieve a sample of 130 refrigerators. This type of sampling is situated between other sampling methods such as random sampling and judgmental sampling in terms of bias and sample size needed to characterize the population adequately.
When considering the provided exercises, the answers to the type of sampling used would be:
The response to a question has three alternatives: A, B, and C. A sample of 120 responses provides 60 A, 23 B, and 37 C. Show the frequency and relative frequency distributions (use nearest whole number for the frequency column and 2 decimal for the relative frequency column).
We know that in a frequency table we write the frequency corresponding to each of the data.
The relative frequency is the ratio of the frequency to the total frequency corresponding to each entry.
Here we have a total sample as: 120
Also, the frequency corresponding to A is: 60
Corresponding to B is: 23
and corresponding to C is: 37
Hence, the Frequency table is as follows:
Data Frequency
A 60
B 23
C 37
The relative frequency table is given by:
Data Relative Frequency
A 60/120=0.5
B 23/120=0.19
C 37/120=0.31
Hence, we get:
Data Relative Frequency
A 0.5
B 0.19
C 0.31
The frequency distribution for the responses A, B, and C are 60, 23, and 37 respectively. The relative frequency distribution for the responses A, B, and C are 0.50, 0.19, and 0.31 respectively. Sum of the relative frequencies equals to 1, indicating all portions of the sample have been accounted for.
Explanation:Frequency and Relative Frequency DistributionsTo create these distributions for the given response data, we'll organize our observations into a table.
Frequency distribution: The frequency shows the number of times each response (A, B, or C) occurs in the sample.
- Response A: 60
- Response B: 23
- Response C: 37
Adding these frequencies would equal the total sample size, 120.
Relative frequency distribution: The relative frequency is calculated as the frequency of each response divided by the total sample size, expressed as a decimal.
- Response A: 60/120 = 0.50
- Response B: 23/120 = 0.19
- Response C: 37/120 = 0.31
Note that if we add up the relative frequencies, we should get a sum of 1, indicating that we've accounted for all portions of the sample.
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Given that y varies directly with x in the table below, what is the value of y if the value of x is 7? x 2 4 6 10 y 12 24 36 60 37 42 48 54
Answer:
42
Step-by-step explanation:
Note that as x increases by 2 from 2 to 4, y increases by 12 from 12 to 24. Thus, the slope of the line connecting the given points is m = rise / run = 12/2, or m = 6. Thus, this direct proportion is written as y = 6x.
If x = 7, y = 6(7) = 42
Answer:
42
Step-by-step explanation:
A soft drink machine outputs a mean of 28 ounces per cup. The machine's output is normally distributed with a standard deviation of 2 ounces. What is the probability of filling a cup between 30 and 31 ounces? Round your answer to four decimal places.
To find the probability of the soft drink machine filling a cup between 30 and 31 ounces, calculate the z scores for 30 and 31 ounces, use them to find the cumulative probabilities from a standard normal distribution table, then subtract the two probabilities. The result is 0.0919 or 9.19%.
Explanation:This is a question about probability in a normal distribution. In this case, we want to find the probability of the output being between 30 and 31 ounces, given a mean of 28 ounces and a standard deviation of 2 ounces.
First, we find the z-scores for 30 and 31 ounces. The z-score is calculated by subtracting the mean from the value and dividing the result by the standard deviation. For 30 ounces, the z-score is (30-28)/2 = 1. For 31 ounces, the z-score is (31-28)/2 = 1.5.
Next, we use these z-scores to find the cumulative probabilities from a standard normal distribution table. The cumulative probability for a z-score of 1 is 0.8413 and for 1.5, it's 0.9332.
The probability of filling a cup between 30 and 31 ounces is the difference between the cumulative probabilities of the two z-scores. So, the answer is 0.9332 - 0.8413 = 0.0919.
Therefore, the probability of the soft drink machine filling a cup between 30 to 31 ounces is 0.0919 or 9.19% when rounded to four decimal places.
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The probability of filling a cup between 30 and 31 ounces is approximately [tex]\(\boxed{0.0919}\)[/tex]
The probability of filling a cup between 30 and 31 ounces when the mean output is 28 ounces and the standard deviation is 2 ounces can be found using the Z-score formula for a normal distribution.
First, we calculate the Z-score for 30 ounces:
[tex]\[ Z_{30} = \frac{X - \mu}{\sigma} = \frac{30 - 28}{2} = \frac{2}{2} = 1 \][/tex]
Next, we calculate the Z-score for 31 ounces:
[tex]\[ Z_{31} = \frac{X - \mu}{\sigma} = \frac{31 - 28}{2} = \frac{3}{2} = 1.5 \][/tex]
Now, we look up the probabilities corresponding to these Z-scores in the standard normal distribution table or use a calculator.
The probability of getting a value less than or equal to [tex]\( Z_{30} \)[/tex]is:
[tex]\[ P(Z \leq 1) \approx 0.8413 \][/tex]
The probability of getting a value less than or equal to is:
[tex]\[ P(Z \leq 1.5) \approx 0.9332 \][/tex]
To find the probability of filling a cup between 30 and 31 ounces, we subtract the probability of filling up to 30 ounces from the probability of filling up to 31 ounces:
[tex]\[ P(30 < X < 31) = P(Z \leq 1.5) - P(Z \leq 1) \][/tex]
[tex]\[ P(30 < X < 31) \approx 0.0919 \][/tex]
Rounded to four decimal places, the probability is 0.0919.
Therefore, the probability of filling a cup between 30 and 31 ounces is approximately [tex]\(\boxed{0.0919}\)[/tex]
In a lottery game, a player picks six numbers from 1 to 48. If 4 of those 6 numbers match those drawn, the player wins third prize. What is the probability of winning this prize?
[tex]|\Omega|={_{48}C_6}=\dfrac{48!}{6!42!}=\dfrac{43\cdot44\cdot45\cdot46\cdot47\cdot48}{720}=12271512\\|A|={_6C_4}=\dfrac{6!}{4!2!}=\dfrac{5\cdot6}{2}=15\\\\P(A)=\dfrac{15}{12271512}=\dfrac{5}{4090504}\approx1.2\%[/tex]
The probability that the player wins the prize from a drawn of six in 1 to 48 numbers is 0.001052.
Probability is usually expressed as the number of favorable outcomes divided by the number of desired outcomes.
Mathematically;
Probability of winning = [tex]\mathbf{\dfrac{favorable \ outcomes}{number \ of \ desired \ outcomes}}[/tex]
The probability that the player wins can be computed by taking the following combinations.
From the given information;
the number of ways in which 4 winning numbers can be selected out of six = [tex]\mathbf{^6C_4}[/tex] the number of ways in which 2 non-winning numbers can be selected out of the 42 non-winning numbers i.e (48 - 6 = 42 ) = [tex]\mathbf{^{42}C_2}[/tex]the number of ways to pick 6 numbers out of 48 = [tex]\mathbf{^{48}C_6}[/tex]Thus, the probability of winning can now be computed as:
[tex]\mathbf{P(winning \ prize) = \dfrac{^6C_4 \times ^{42}C_2}{^{48}C_6}}[/tex]
[tex]\mathbf{P(winning \ prize) = \dfrac{\Big (\dfrac{6!}{4!(6-4)!} \Big )\times \Big (\dfrac{42!}{2!(42-2)!} \Big )}{\Big (\dfrac{48!}{6!(48-6)!} \Big )}}[/tex]
[tex]\mathbf{P(winning \ prize) = \dfrac{12915}{12271512}}[/tex]
[tex]\mathbf{P(winning \ prize) =0.001052}[/tex]
Therefore, the probability that the player wins the prize from a drawn of six in 1 to 48 numbers is 0.001052.
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At a local college, 102 of the male students are smokers and 408 are non-smokers. Of the female students, 240 are smokers and 360 are non-smokers. A male student and a female student from the college are randomly selected for a survey. What is the probability that both are non-smokers? Do not round your answer.
Answer: Probability that both are non smokers is 0.48.
Step-by-step explanation:
Since we have given that
Number of male smokers = 102
Number of male non smokers = 408
Total male = 102+408=510
Number of female smokers = 240
Number of female non smokers = 360
Total female = 240+360 = 600
According to question,
A male student and a female student from the college are randomly selected for a survey,
So, Probability that both are non smokers is given by
P(both are smokers ) = P(Male smoker) × P(female smoker)
[tex]P(both)=\dfrac{360}{600}\times \dfrac{408}{510}\\\\P(both)=\dfrac{146880}{306000}\\\\P(both)=0.48[/tex]
Hence, probability that both are non smokers is 0.48.
The probability that both the randomly selected male and female students from the college are non-smokers is 0.48.
Explanation:To compute the probability that both the randomly selected male and female students are non-smokers, we start by finding the total number of male and female students. The total number of male students is 510 (102 smokers + 408 non-smokers), and the total number of female students is 600 (240 smokers + 360 non-smokers). Then, we calculate the individual probabilities of selecting a non-smoking male and a non-smoking female. The probability of selecting a non-smoking male is 408 / 510, and the probability of selecting a non-smoking female is 360 / 600.
To find the combined probability, we multiply these individual probabilities. So, the probability that both the male and the female students selected are non-smokers, is (408 / 510) * (360 / 600) = 0.8 * 0.6 = 0.48.
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Part A: Sam rented a boat at $225 for 2 days. If he rents the same boat for 5 days, he has to pay a total rent of $480.
Write an equation in the standard form to represent the total rent (y) that Sam has to pay for renting the boat for x days. (4 points)
Answer:
y = 85x + 55
Step-by-step explanation:
If we use the given prices as coordinates we will get the coordinates (2, 225) and (5, 480) and if we use the rise over run which in this case is 255/3 then we get the slope which is $85 and now we plug what we know into y = mx + b. we now have the equation 225 = (85)2 + b which can be simplified to 55 = b and now we can create the equation y = 85x + 55.
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Q1. Ten percent of the items produced by a machine are defective. A random sample of 100 items is selected and checked for defects. What is the standard error of the distribution of the sample proportions? Round your answer to three decimal places.
Answer: 0.030
Step-by-step explanation:
Given: The percent of the items produced by a machine are defective :[tex]P=10\%=0.10[/tex]
The percent of the items produced by machine which are not defective:[tex]Q=1-0.1=0.9[/tex]
Sample size : n = 100
Now, the standard error of the proportion is given by :-
[tex]\text{S.E.}=\sqrt{\dfrac{0.1\times0.9}{100}}\\\\\Rightarrow\text{S.E.}=0.030[/tex]
Hence, the the standard error of the distribution of the sample proportions=0.030
The standard error for the distribution of sample proportions of a machine producing 10% defective items determined from a random sample of 100 items is ±0.030.
Explanation:Ten percent of the items produced by a machine are defective. From a random sample of 100 items, the standard error of the sample proportions is calculated using the formula for the standard error of a proportion, which is √(pq/n), where p is the proportion of successes, q is the proportion of failures (1-p), and n is the sample size.
In this case, p is 0.10 (the proportion defective), q is 0.90 (1-p, representing the proportion not defective), and n is 100 (the sample size). Substituting these values into the formula gives: √[(0.10)(0.90) / 100] which equals ±0.030 (rounded to three decimal places).
Therefore, the standard error for the distribution of sample proportions is ±0.030.
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