Answer:
0.266
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 5.85, \sigma = 0.24[/tex]
Find the probability that a randomly selected mandarin orange from this farm has a diameter larger than 6.0 cm.
This is 1 subtracted by the pvalue of Z when X = 6.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{6 - 5.85}{0.24}[/tex]
[tex]Z = 0.625[/tex]
[tex]Z = 0.625[/tex] has a pvalue of 0.734
1 - 0.734 = 0.266
Answer:
[tex]P(X>6)=P(\frac{X-\mu}{\sigma}>\frac{6-\mu}{\sigma})=P(Z>\frac{6-5.85}{0.24})=P(z>0.625)[/tex]
And we can find this probability using the complement rule and the normal standard table or excel:
[tex]P(z>0.625)=1-P(z<0.625)=1-0.734= 0.266[/tex]
Step-by-step explanation:
Previous concepts
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".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the diameters of mandarin oranges of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(5.85,0.24)[/tex]
Where [tex]\mu=5.85[/tex] and [tex]\sigma=0.24[/tex]
We are interested on this probability
[tex]P(X>6)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(X>6)=P(\frac{X-\mu}{\sigma}>\frac{6-\mu}{\sigma})=P(Z>\frac{6-5.85}{0.24})=P(z>0.625)[/tex]
And we can find this probability using the complement rule and the normal standard table or excel:
[tex]P(z>0.625)=1-P(z<0.625)=1-0.734= 0.266[/tex]
identify and interpret (explain) one other point on the graph.?
Answer:
can i getttt a pictture
Step-by-step explanation:
Find the diameter of the circle with the given circumference use 3.14 C=18
A real-valued function f is said to be periodic with period T ≠ 0 if f(x + T) = f(x) for all x in the domain of f. If T is the smallest positive value for which f(x + T) = f(x) holds, then T is called the fundamental period of f. Determine the fundamental period T of the given function. f(x) = sin(2x) + cos(4x)
Answer:
Period T of the given function f(x) = sin(2x) + cos(4x)
= π
Step-by-step explanation:
Given that y(x) is a sum of two trigonometric functions. The period T of sin 2x would be (2π÷2) = π. Period T of cos4x would be (2π÷4) that is π/2
Find the LCM of π and π/2 . That would be π. Hence the period of the given function would be π
The fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \) is:\[ T = \pi \][/tex].
To determine the fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \)[/tex], we need to find the smallest positive value of ( T ) such that [tex]f(x + T) = f(x) \) for all \( x \).\\[/tex]
Let's start by analyzing the periods of the individual components of [tex]\( f(x) \).[/tex]
1. Period of [tex]\( \sin(2x) \):[/tex]
The standard period of [tex]\( \sin(x) \) is \( 2\pi \). For \( \sin(2x) \),[/tex] the argument ( 2x ) scales the period. To find the period of [tex]\( \sin(2x) \)[/tex], we set:
[tex]\[ 2x = 2x + 2\pi \]\\[/tex]
Solving for the period, we get:
[tex]\[ x = x + \frac{2\pi}{2} \][/tex]
Thus, the period of [tex]\( \sin(2x) \)[/tex] is:
[tex]\[ \frac{2\pi}{2} = \pi \][/tex]
2. Period of [tex]\( \cos(4x) \):[/tex]
The standard period of [tex]\( \cos(x) \) is \( 2\pi \). For \( \cos(4x) \)[/tex], the argument ( 4x ) scales the period. To find the period of [tex]\( \cos(4x) \),[/tex] we set:
[tex]\[ 4x = 4x + 2\pi \][/tex]
Solving for the period, we get:
[tex]\[ x = x + \frac{2\pi}{4} \][/tex]
Thus, the period of [tex]\( \cos(4x) \)[/tex]is:
[tex]\[ \frac{2\pi}{4} = \frac{\pi}{2} \][/tex]
Next, we need to find the smallest positive ( T ) such that both [tex]\( \sin(2x) \)[/tex] and [tex]\( \cos(4x) \)[/tex] have the same period ( T ). This means that ( T ) must be a common multiple of the periods of the two components, [tex]\( \pi \)[/tex] and [tex]\( \frac{\pi}{2} \).[/tex]
To find the fundamental period ( T ), we determine the least common multiple (LCM) of [tex]\( \pi \) and \( \frac{\pi}{2} \):[/tex]
- [tex]\( \pi \)[/tex]can be written as [tex]\( \pi \times 1 \).[/tex]
- [tex]\( \frac{\pi}{2} \)[/tex] can be written as [tex]\( \pi \times \frac{1}{2} \).[/tex]
The LCM of [tex]\( 1 \) and \( \frac{1}{2} \) is \( 1 \) since \( 1 \)[/tex] is the smallest number that both [tex]\( 1 \) and \( \frac{1}{2} \)[/tex] can divide without leaving a remainder.
Thus, the LCM of [tex]\( \pi \) and \( \frac{\pi}{2} \) is:[/tex]
[tex]\[ \text{LCM}\left(\pi, \frac{\pi}{2}\right) = \pi \][/tex]
Therefore, the fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \) is:\[ T = \pi \][/tex]
Order the decimals from least to greatest 7.508,5.2161,7.5,7.58
Answer:
5.2161,7.5,7.508,7.58
Step-by-step explanation:
Answer:
5.2161; 7.5; 7.508; 7.58
Step-by-step explanation:
-First you will look at the first digit of each number given, obviously 5 is the smallest so it will go first
-Then you will check the following number after the first digit to see which one is smaller of the "7" values. In this case, they all have the number "5" has their second value, so we move on to the third digit.
-Since "7.5" does not have a third digit, it will be the smallest of the number "7" values.
-We then have a remaining of two values, 7.508 and 7.58. Still looking at the third numbers of each of the two, we see "0" and "8," and obviously 0 is smaller than 8, so 7.508 is smaller than 7.58
Select each polynomial that is a perfect square.
Question 5 options:
x2−10x−25
4x2−12x+9
9x2+12x+16
16x2+16x+1
Answer:
4x^2−12x+9
Step-by-step explanation:
The form of a perfect square trinomial is ...
(a +b)² = a² +2ab +b²
The first and last terms must be positive and perfect squares. The middle term must be twice the product of their roots (possibly with a minus sign).
x^2 -10x -25 . . . . -25 is not a positive perfect square
4x^2 -12x +9 . . . . 12x = 2√(4x^2·9) = 2·6x . . . . perfect square
9x^2 +12x +16 . . . 12x ≠ 2√(9x^2·16) = 2·12x
16x^2 +16x +1 . . . 16x ≠ 2√(16x^2·1) = 2·4x
Suppose that on a true/false exam you have no idea at all about the answers to three questions. You choose answers randomly and therefore have a 50–50 chance of being correct on any one question. Let CCW indicate that you were correct on the first two questions and wrong on the third, let WCW indicate that you were wrong on the first and third questions and correct on the second, and so forth. a. List the elements in the sample space whose outcomes are all possible sequences of correct and incorrect responses on your part. b. Write each of the following events as a set and find its probability: (i) The event that exactly one answer is correct. (ii) The event that at least two answers are correct. (iii) The event that no answer is correct.
Answer:
a.
[tex]\text{Sample Space} = \{ CCC,CCW,CWW,WWW,WWC,WCC,WCW,CWC \}[/tex]
b.
(i) 1/2
(ii) 2/3
(iii) 1/6
Step-by-step explanation:
a.
The sample space is the list of all possibilities.
[tex]\text{Sample Space} = \{ CCC,CCW,CWW,WWW,WWC,WCC,WCW,CWC \}[/tex]
b.
(i)
If exactly one answer is correct the favorable outcomes are
CWW , WCW , WWC.
And the probability would be 3/6 = 1/2.
(ii)
If at least two answers are correct then the favorable outcomes are
CCC,CCW,WCC,CWC
and the probability is 4/6 = 2/3.
(iii)
If no answer is correct, the favorable outcomes are
WWW
and the probability is 1/6.
store sold combined totalof 412 biology and math books in a week. the number of biology textbooks sold was three times the number of math books sold. how many textbooks of each type were sold?
Step-by-step explanation:
Total books sold = 412
So if there are 103 maths books then biology books should be three times = 103 × 3 = 309
So when we add 103 + 309 = 412
It means the number of biology books is 309 and maths book is 103
If you put $6.57 into a savings account that earns 4%, how much interest
will you receive at the end of eight years?
The interest for 8 years is $2.10
Step-by-step explanation:
Principal amount (p) = $6.57
Rate of interest (r) = 4%
Time (t) = 8 years
Interest = (p x r x t) /100
= (6.57 x 4 x 8) /100
= 210.24/100
= 2.10
The interest for 8 years is $2.10
Much like sound bytes of news stories, statistical studies are often reduced to one- or two- sentence stat-bytes. For the following stat-byte, discuss what crucial information is missing and what more you would want to know before acting on the study. A cable network reports on a survey of America's top restaurants that found that "only nine restaurants achieved a rare 29 out of a possible 30 rating and none of those restaurants are in the Big Apple." Which of the following are crucial information that you would want to know before acting on the study? Select all that apply.
A. The goal of the study
B. How the quality of restaurants was measured
C. The variable of interest
D. Who the respondents in the survey were
E. How the respondents were selected
To make informed decisions based on a statistical study about restaurants, it's critical to understand the goal of the study, how the quality was measured, what the variable of interest was, who the respondents were and how they were selected.
Explanation:The crucial information that would need to know before acting on the study mentioned in the question includes all of the options provided and more. Let's discuss each option in detail:
A. The goal of the study. This would provide context and help to determine why these specific variables were chosen and measured. Without a clear goal, interpreting findings can be difficult.
B. How the quality of restaurants was measured. It's important to understand how the '29 out of 30' rating was developed - what specific factors led to this score. If the rating system or measurement techniques are flawed, the results could possibly be inaccurate.
C. The variable of interest. It would be necessary to understand what variable is being measured. Is it the food quality, service, ambiance, or some other factor of a restaurant's operation?
D. Who the respondents in the survey were. The demographic and social details of the respondents are also vital. If all respondents are from a certain group (for example, high-income people), results may not be representative for the general public.
E. How the respondents were selected. This would throw light on the methodology and the degree to which the findings can be generalized to the larger population.
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Multiply the polynomials. (2x+1)(x−5)
Answer:
2x^2-9x-5
Step-by-step explanation:
Just multiply
Answer:
2x^2-9x-5
Step-by-step explanataion:
multiply
A bread recipe calls for 1 teaspoon of yeast for every 2
cups of flour.
Write an equation that represents the number of cups of
flour, c, for every teaspoon of yeast, t.
Equation representing the cups of flour for every teaspoon of yeast will be → c = 2t
Proportional relation between two variables: If a variable 'y' is directly proportional to another variable 'x', expression will be,
y ∝ x
y = kx [Here, k = proportionality constant]
It has been given in the question,
"A bread recipe calls for 1 teaspoon of yeast for every 2 cups of flour."
If the number of teaspoons of yeast is represented by 't' and number of cups of flour by 'c',
c = kt
From the given statement,
2 = k × 1
k = 2
Therefore, equation for the given statement will be → c = 2t
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Find two positive numbers such that the ratio of the two numbers is 7 to 3 and the product of the two numbers is 525.
Answer:
35 and 15
Step-by-step explanation:
Process of elimination:
Try 105 and 5 - 105 is way too much and 5 is way too little. We know that they are between 5 and 105.
Try 50 and 10.5 - This is closer, but 50 is still too much and 10.5 is too little. We now know that the two numbers are between 10.5 and 50
Try 30 and 17.5 - This is very close, but now 30 is not enough and 17.5 is too much. We now know that the smaller number is between 10.5 and 17.5 as well as the larger number is between 30 and 50.
Try 40 and 13.125 - This is also very close, but 40 is too much and 13.125 is too little. We now know that the lower number is between 13.125 and 17.5 as well as the larger number is between 30 and 40.
Try 35 and 15 - Success: 35 and 5 make the simplified ratio of 7:3! They also are products of 525!
Final answer:
The two positive numbers with a ratio of 7 to 3 and a product of 525 are found to be 35 and 15, by assigning a constant to the numbers, forming and solving a quadratic equation to find the value of the constant, and then using it to find the two numbers.
Explanation:
To find two positive numbers where their ratio is 7 to 3, and their product is 525, we can assign values x and y to the numbers such that x/y = 7/3, and x*y = 525. We can solve these equations simultaneously to find the values of x and y.
Step-by-step solution:
Let the two numbers be 7k and 3k, where k is a constant. This ensures that the ratio 7 to 3 is preserved.Since the product of the two numbers is 525, we have that 7k * 3k = 525.Combining like terms, we get 21k^2 = 525.Dividing both sides by 21 gives k^2 = 25.Taking the square root of both sides, we find that k = 5.Substitute k = 5 back into 7k and 3k to find the two numbers: 7*5 = 35 and 3*5 = 15.Therefore, the two numbers are 35 and 15.
The image of the lemon is at point I. What is the size of the image compared to the size of the lemon?
same size...hope this helps :)))))
Find the equation of a quadratic function from its graph|
Answer:
U shape
Step-by-step explanation: look for a U shape and youll have chosen a quadratic function but find the vertex and put them into the format then find the slope
Which operation is performed in the derivation of the quadratic formula moving from Step 6 to Step 7? subtracting StartFraction b Over 2 a EndFraction from both sides of the equation squaring both sides of the equation taking the square root of both sides of the equation taking the square root of the discriminant
Answer: C. Taking the square root of both sides of the equation.
Step-by-step explanation:
Answer:
c
Step-by-step explanation:
i got it right
A rePrism measures 6.7 cm by 3.2 cm by 9 cm what is the volume of the rectangle prism
Answer:
The volume is 192.96
Step-by-step explanation:
You multiply length x width x height to find the volume.
The radius of the large sphere is times longer than the radius of the small sphere.
How many times the volume of the large sphere is the volume of the small sphere?
A-1/27
B-1/18
C-1/9
D-1/3
Answer:
1/3
Step-by-step explanation:
You can try it I keep getting 3 so maybe that’s the answer
1/3, because 18.6 / 3 = 6.2
Suppose Julio is a veterinarian who is doing research into the weight of domestic cats in his city. He collects information on 131 cats and finds the mean weight for cats in his sample is 10.87 lb with a standard deviation of 4.31 lb. What is the estimate of the standard error of the mean (SE)
Answer:
Estimate of the standard error of the mean = 0.38 lb
Step-by-step explanation:
We are given the following in the question:
Sample mean, [tex]\bar{x}[/tex] = 10.87 lb
Sample size, n = 131
Standard deviation, σ = 4.31 lb
We have ti find the estimate of the standard error of the mean.
Formula for standard error:
[tex]S.E= \dfrac{\sigma}{\sqrt{n}}[/tex]
Putting values, we get,
[tex]S.E = \dfrac{4.31}{\sqrt{131}} = 0.3766 \approx 0.38[/tex]
0.38 lb is the standard error of the mean.
Solve the division problem. Round answer to the nearest hundredth.
9.2.15 2.0 6 3
Answer:
6
Step-by-step explanation:
change the decimal number 3.5 into a mixed number
Answer: [tex]3\frac{1}{2}[/tex]
Multiply
[tex]3.5/1*10/10=35/10[/tex]
Divide each side by 5
[tex]35/10[/tex] ÷ [tex]5/5=7/2[/tex]
[tex]7/2=3\frac{1}{2}[/tex]
Answer:
[tex] \frac{3.5}{1} \frac{ \times }{ \times } \frac{10}{10} = \frac{35}{10} \\ \frac{35}{10} \frac{ \div }{ \div } \frac{5}{5} \\ = \frac{7}{2} = 3 \frac{1}{2} [/tex]
hope this helps you...
Please hurry i don't have much time left!
If this rectangle is dilated using a scale factor of One-half through point B, what is the result?
Answer:
Check below
Step-by-step explanation:
That's too bad you haven't attached a rectangle.
Here's an example, with the data you've typed in.
1) When we dilate a rectangle we either grows it or shrink it through a scale factor.
Check the first picture below.
The New Dilated Rectangle A'B'C'D' will follow its coordinates, when the Center of Dilation is at its origin(Middle):
[tex]D_{A'B'C'D'}=\frac{1}{2}(x,y)[/tex]
2) But In this question, B is the center of Dilation. So, Since B is the Dilation Point B=B' . And More importantly:
[tex]\bar{AB}=\frac{1}{2}\bar{A'B'}\\\bar{CD}=\frac{1}{2}\bar{C'D'}\\\bar{AC}=\frac{1}{2}\bar{A'C'}\\\bar{CD}=\frac{1}{2}\bar{C'D'}\\[/tex]
3) So check the pictures below for a better understanding.
Answer:
Pretty sure its B
1. Solve for x.
Need help in math
Answer:
21 maybe?
Step-by-step explanation:
if those numbers represent lengths and those lines are the same size, x is 21
Greg Evaluated has And found that he was spending $50 more per month on Utilities that he has budgeted. He can transfer money from other categories To increase his utility budget to $125 per month. If his total monthly income is $2400 To the nearest percent, what Percent of his monthly income will be budgeted for utilities?
Answer:
22%
Step-by-step explanation:
Answer:
5%
Step-by-step explanation:
There are two ways to solve this
a fast way: 125/2400 which equals approx 0.05208
when you convert this to a percentage it is about 5%
another way is take 2400 has 100% and 125 as x%
so
125/2400 = x/100
cross multiply so
2400x = 12500
then
x= 12500/2400 which simplifies to about
5.208. since this was out of 100 from the start you don't need to convert it to percentage form
The length of the rectangle is 4 less than twice the width . The perimeter is 28 . Find the length and width .
Answer:
Length=8 and width=6
Step-by-step explanation:
The length of the rectangle is 4 less than twice the width, let l stand for length and w for width: l=2w-4
The perimeter is equal to 28 and the formula for the perimeter of a rectangle is 2l+2w: 28=2l+2w
Substitute l into the equation:
28=2(2w-4)+2w
28=4w-8+2w
28=6w-8
36=6w
w=6
Find for l:
l=2w-4
l=2(6)-4
l=12-4
l=8
Use the normal distribution to find a confidence interval for a proportion p given the relevant sample results. Give the best point estimate for p, the margin of error, and the confidence interval. Assume the results come from a random sample. A 99% confidence interval for p given that p-hat = 0.34 and n= 500. Point estimate ___________ (2 decimal places) Margin of error __________ (3 decimal places) The 99% confidence interval is ________ to _______ (3 decimal places)
Answer:
(a) The point estimate for the population proportion p is 0.34.
(b) The margin of error for the 99% confidence interval of population proportion p is 0.055.
(c) The 99% confidence interval of population proportion p is (0.285, 0.395).
Step-by-step explanation:
A point estimate of a parameter (population) is a distinct value used for the estimation the parameter (population). For instance, the sample mean [tex]\bar x[/tex] is a point estimate of the population mean μ.
Similarly, the the point estimate of the population proportion of a characteristic, p is the sample proportion [tex]\hat p[/tex].
The (1 - α)% confidence interval for the population proportion p is:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
The margin of error for this interval is:
[tex]MOE= z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
The information provided is:
[tex]\hat p=0.34\\n=500\\(1-\alpha)\%=99\%[/tex]
(a)
Compute the point estimate for the population proportion p as follows:
Point estimate of p = [tex]\hat p[/tex] = 0.34
Thus, the point estimate for the population proportion p is 0.34.
(b)
The critical value of z for 99% confidence level is:
[tex]z={\alpha/2}=z_{0.01/2}=z_{0.005}=2.58[/tex]
*Use a z-table for the value.
Compute the margin of error for the 99% confidence interval of population proportion p as follows:
[tex]MOE= z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
[tex]=2.58\sqrt{\frac{0.34(1-0.34)}{500}}[/tex]
[tex]=2.58\times 0.0212\\=0.055[/tex]
Thus, the margin of error for the 99% confidence interval of population proportion p is 0.055.
(c)
Compute the 99% confidence interval of population proportion p as follows:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
[tex]CI=\hat p\pm MOE[/tex]
[tex]=0.34\pm 0.055\\=(0.285, 0.395)[/tex]
Thus, the 99% confidence interval of population proportion p is (0.285, 0.395).
The point estimate for p is 0.34. The margin of error, calculated using a z-score of 2.576, is 0.034. The 99% confidence interval is from 0.306 to 0.374.
Explanation:This question is about calculating a confidence interval for a proportion using the normal distribution. The best point estimate for p is the sample proportion, p-hat, which is 0.34.
For a 99% confidence interval, we use a z-score of 2.576, which corresponds to the 99% confidence level in a standard normal distribution. The formula for the margin of error (E) is: E = Z * sqrt[(p-hat(1 - p-hat))/n]. Substituting into the formula, E = 2.576 * sqrt[(0.34(1 - 0.34))/500] = 0.034.
The 99% confidence interval for p is calculated by subtracting and adding the margin of error from the point estimate: (p-hat - E, p-hat + E). The 99% confidence interval is (0.34 - 0.034, 0.34 + 0.034) = (0.306, 0.374).
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How can you best describe a stop sign using
polygons?
The sign has 8 sides, so it is an octagon
It appears to be
because the sides and
angles appear to be congruent.
STOP
Dene
Answer:
The answer to the last one is regular
Step-by-step explanation:
A polygon is a planar figure defined by a finite number of straight-line segments. The polygon that best describes a stop sign is a regular polygon.
What is a polygon?A polygon is a planar figure defined by a finite number of straight-line segments that are joined to form a closed polygonal chain in geometry. A polygon can be defined as a bounded planar region, a bounding circuit, or both.
The polygon that best describes a stop sign is a regular octagon because a stop sign has 8 sides and the length of each side is the same, therefore, the polygon will be a regular polygon.
Thus, the polygon that best describes a stop sign is a regular polygon.
Learn more about Polygon:
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For the 405 highway that car pass through a checkpoint, assume the speeds are normally distributed such that μ= 61 miles per hour and δ=4 miles per hour. Calculate the Z value for the next car that passes through the checkpoint will be traveling slower than 65 miles per hour.
Answer:
[tex]Z = 1[/tex]
Step-by-step explanation:
Z - score
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 61, \sigma = 4[/tex]
Calculate the Z value for the next car that passes through the checkpoint will be traveling slower than 65 miles per hour.
This is Z when X = 65. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{65 - 61}{4}[/tex]
[tex]Z = 1[/tex]
Find the Laplace transform of the given function: f(t)=(t−5)u2(t)−(t−2)u5(t), where uc(t) denotes the Heaviside function, which is 0 for t
[tex]f(t)=(t-5)u_2(t)-(t-2)u_5(t)[/tex]
The Laplace transform is
[tex]F(s)=\displaystyle\int_0^\infty f(t)e^{-st}\,\mathrm dt=\int_2^\infty(t-5)e^{-st}\,\mathrm dt-\int_5^\infty(t-2)e^{-st}\,\mathrm dt[/tex]
Integrate by parts; in the first integral, take
[tex]u=t-5\implies\mathrm du=\mathrm dt[/tex]
[tex]\mathrm dv=e^{-st}\,\mathrm dt\implies v=-\dfrac{e^{-st}}s[/tex]
[tex]\implies\displaystyle\int_2^\infty(t-5)e^{-st}\,\mathrm dt=-\frac{e^{-st}}s(t-5)\bigg|_2^\infty+\frac1s\int_2^\infty e^{-st}\,\mathrm dt[/tex]
[tex]=-\dfrac{3e^{-2s}}s-\dfrac{e^{-st}}{s^2}\bigg|_2^\infty[/tex]
[tex]=-\dfrac{3e^{-2s}}s+\dfrac{e^{-2s}}{s^2}=-\dfrac{(3s-1)e^{-2s}}{s^2}[/tex]
For the second integral, take
[tex]u=t-2\implies\mathrm du=\mathrm dt[/tex]
[tex]\mathrm dv=e^{-st}\,\mathrm dt\implies v=-\dfrac{e^{-st}}s[/tex]
[tex]\implies\displaystyle\int_5^\infty(t-2)e^{-st}\,\mathrm dt=-\dfrac{(t-2)e^{-st}}s\bigg|_5^\infty+\frac1s\int_5^\infty e^{-st}\,\mathrm dt[/tex]
[tex]=\dfrac{3e^{-5s}}s-\dfrac{e^{-st}}{s^2}\bigg|_5^\infty[/tex]
[tex]=\dfrac{3e^{-5s}}s+\dfrac{e^{-5s}}{s^2}=\dfrac{(3s+1)e^{-5s}}{s^2}[/tex]
So we have
[tex]F(s)=\dfrac{(3s+1)e^{-5s}-(3s-1)e^{-2s}}{s^2}[/tex]
The Shady Farm Milk Company can process milk at a fixed rate of 7500 gallons/hour. The company’s clients request 100,000 gallons of milk over the course of one day. This demand is spread out uniformly from 8 a.m. to 6 p.m. The company starts producing at 8 a.m. and continues to work until all of the demand has been satisfied. At noon, how many gallons of milk are in the queue to be processed?
At noon, the Shady Farm Milk Company has 10,000 gallons of milk in the queue to be processed given the demand and the processing rate.
Explanation:The Shady Farm Milk Company can process 7500 gallons of milk per hour. Given that the company operates from 8 a.m. to 6 p.m., this is a total of 10 hours of operation in a day. Therefore, in 10 hours, the company can process 7500 × 10 = 75,000 gallons of milk.
However, the demand for milk is 100,000 gallons over the course of the day. Therefore, by noon, the company has been operating for 4 hours, meaning they can process 7500 × 4 = 30,000 gallons.
The demand over the same 4 hours period (from 8 a.m. to noon) is calculated by dividing the total demand over the entire course of the day (which is evenly spread) by the number of operating hours. Thus: 100,000 / 10 = 10,000 gallons/hour.
Consequently, the demand from 8 a.m. to noon is: 10,000 × 4 = 40,000 gallons. So, the amount of milk in the queue at noon would be the demand minus what the company has processed at that time.
Hence: 40,000 (demand from 8 a.m. to noon) - 30,000 (processed milk from 8 a.m. to noon) = 10,000 gallons. Therefore, at noon, the company has 10,000 gallons of milk in the queue to be processed.
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At noon, there are 20,000 gallons of milk in the queue to be processed.
To find out how many gallons of milk are in the queue to be processed at noon, we first need to calculate how many gallons of milk have been processed by noon.
The company can process milk at a fixed rate of 7500 gallons per hour. From 8 a.m. to noon, there are 4 hours.
Total gallons processed by noon = Rate of processing Time
[tex]\[ = 7500 \, \text{gallons/hour} \times 4 \, \text{hours} = 30000 \, \text{gallons} \][/tex]
Now, we need to find out how many gallons of milk are still in demand by noon. The total demand over the course of the day is 100,000 gallons, and it is spread out uniformly from 8 a.m. to 6 p.m.
This means that by noon, half of the day has passed.
So, the total demand by noon = Total demand / 2
[tex]\[ = \frac{100000}{2} = 50000 \, \text{gallons} \][/tex]
Now, to find out how many gallons are in the queue to be processed at noon, we subtract the gallons already processed from the total demand:
Gallons in the queue at noon = Total demand by noon - Gallons processed by noon
[tex]\[ = 50000 \, \text{gallons} - 30000 \, \text{gallons} = 20000 \, \text{gallons} \][/tex]
So, at noon, there are 20,000 gallons of milk in the queue to be processed.
Which expression could be used to determine the area of the triangle shown? One-half + 12 and one-third + 3 One-half (3) (12) (one-third) 12 and one-third times 3 One-half (12 and one-third) (3)
Answer:
The answer is D.
Step-by-step explanation: