Suppose an airline policy states that all baggage must be box shaped with a sum of length, width, and height not exceeding 96 in. What are the dimensions and volume of a square-based box with the greatest volume under these conditions?
Answers
Answer:
Step-by-step explanation:
As the base is a square so the length is a, width is a and the height is h.
According to the question,
a + a + h = 96
h = 96 - 2a .... (1)
Volume of the box, V = length x width x height
V = a x a x h
V = a² (96 - 2a) from equation (1)
V = 96a² - 2a³
Differentiate both sides
Now put it equal to zero.
192 a - 6a² = 0
a = 32 in
h = 96 - 2 x 32
h = 32 in
Thus, the length and the width os teh base is 32 in and the height is 32 in.
Final answer:
A square-based box with the greatest volume under a restriction of the sum of length, width, and height not exceeding 96 inches must have each dimension equal to 32 inches. Therefore, its volume will be 32x32x32 = 32,768 cubic inches.
Explanation:
A square-based box with the greatest volume that can fit the airline's restrictions would have each side (length, width, height) be exactly one third of the total permitted sum, namely 32 inches each, because the volume of a square-based box (a cube in this specific case) is calculated by cubing the edge length. This is due to the nature of a cube, where all sides are equal.
So, the volume of the box would be 32in x 32in x 32in = 32,768 cubic inches. This is the maximum volume because the mathematical principle that for a given sum S of width, length & height, a cube always takes up the most volume.
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Related Questions
A quality-control manager for a company that produces a certain soft drink wants to determine if a 12-ounce can of a certain brand of soft drink contains 120 calories as the labeling indicates. Using a random sample of 10 cans, the manager determined that the average calories per can is 124 with a standard deviation of 6 calories. At the .05 level of significance, is there sufficient evidence that the average calorie content of a 12-ounce can is greater than 120 calories? Assume that the number of calories per can is normally distributed.
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Taylor charges $15 for the first hour of dog walking and $10 for each additionalhour or fraction of an hour. The rates that Lucy charges for x hours of dog walking are modeled with the function shown.f(x) ={20x, 010x. 2 5x, x >4Who will charge more to walk a dog for 2.5 hours, Taylor or Lucy? How much more?Who?:How much more?:
Answers
Answer:
Sampling error
Step-by-step explanation:
The answer is sampling error.
The sampling error occurs when the sample does not represent the full population and the result from the sample is not a representation of the results from full sample.
From information given
Population mean = 87.85
Population standard deviation = 118.1
n = 25
Sample mean = 79.07
Sample standard deviation = 129.91
118.2/√25
= 23.62
The standard deviation of the distribution is what is referred to as standard error of m = 23.62
Final answer:
The difference between the population mean and sample mean is likely due to sampling variability, a concept related to the Central Limit Theorem. Given the small sample size in this case, it's not unusual to see this difference.
Explanation:
The difference between the mean of the population (μ) and the mean of the sample (M) is likely attributable to the phenomenon known as sampling variability. This is a concept central to the Central Limit Theorem, which states that when enough random samples are taken from a population, the distribution of the means of these samples will approximate a normal distribution, even if the original population distribution is not normal. The mean of this distribution will be equal to the populating mean, and its standard deviation will be the standard deviation of the population, divided by the square root of the sample size (n).
In this specific case, you've taken a relatively small sample (n=25) from a larger population (N=2,431). Consequently, it is not unexpected that there is some difference between the population mean (μ = 87.85) and the sample mean (M = 79.07). However, as you increase the number of samples you are drawing, according to the Central Limit Theorem, the average of these sample means should converge on the population mean.
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Answers
Answer:
We can therefore conclude that the geographical distribution of hotline callers could be the same as the U.S population distribution.
Step-by-step explanation:
The null Hypothesis: Geographical distribution of hotline callers could be the same as the U.S. population distribution
Alternative hypothesis: Geographical distribution of hotline callers could not be the same as the U.S. population distribution
The populations considered are the Midwest, South, Northeast, and west.
The number of categories, k = 4
Number of recent calls = 200
Let the number of estimated parameters that must be estimated, m = 0
The degree of freedom is given by the formula:
df = k - 1-m
df = 4 -1 - 0 = 3
Let the significance level be, α = 5% = 0.05
For α = 0.05, and df = 3,
from the chi square distribution table, the critical value = 7.815
Observed and expected frequencies of calls for each of the region:
Northeast
Observed frequency = 39
It contains 18.1% of the US Population
The probability = 0.181
Expected frequency of call = 0.181 * 200 = 36.2
Midwest
Observed frequency = 55
It contains 21.9% of the US Population
The probability = 0.219
Expected frequency of call = 0.219 * 200 =43.8
South
Observed frequency = 60
It contains 36.7% of the US Population
The probability = 0.367
Expected frequency of call = 0.367 * 200 = 73.4
West
Observed frequency = 46
It contains 23.3% of the US Population
The probability = 0.233
Expected frequency of call = 0.233 * 200 = 46
Where observed frequency
Expected frequency
Calculate the test statistic value, x²
Since the test statistic value, x²= 5.535 is less than the critical value = 7.815, the null hypothesis will not be rejected, i.e. it will be accepted. We can therefore conclude that the geographical distribution of hotline callers could be the same as the U.S population distribution.
Answers
1.8 that is the best answer
Answer:
0.56
Step-by-step explanation:
A quotient is a quantity produced by the division of two numbers.
5 / 9 = 0.555555556
0.555555556 = 0.56
image attached
Answers
Answer:
average rate of change = - 0.5
Step-by-step explanation:
the average rate of change of f(x) in the closed interval [ a, b ] is
here the closed interval is [ 1, 3 ] , then
f(b) = f(3) = 4 ← point (3, 4 ) on graph
f(a) = f(1) = 5 ← point (1, 5 ) on graph
Then
average rate of change = =
= - 0.5
Answers
Answer:
Step-by-step explanation:
Hello!
The objective of this exercise is to test if the Y: "number of car deaths in one month" is affected by the variable X: "seat belt law"
The linear regression was estimated:
Coefficients: Estimate Std. Error t value Pr(> | t |)
(Intercept) 125.870 1.849 68.082 < 2e-16 *
Seatbelts -25.609 5.342 -4.794 3.29e-06 *
R-squared = 0.11
Then the estimated model is:
Yi= 125.870 - 25609Xi
a. Did the seat belt law make a difference?
Yes.
If the hypothesis is that the seat belt law reduces the number of car deaths:
H₀: β ≥ 0
H₁: β < 0
With α: 0.05
The p-value for the test is: 3.29e-06
The p-value is less than the significance level, the seat belt law modifies the average number of car deaths.
b. Is there a need to add more variables to the model?
Yes. According to the given model, the independent variable isn't good enough to explain the variability of the dependent variable, i.e. most of the variability of the dependent variable is given by the errors.
The investigator needs to add new variables or change the model to determine one that is a better predictor of the dependent variable.
c. How would you justify your answer with numbers?
To see if the independent variable is a good predictor of the dependent variable you have to look at the coefficient of determination. This coefficient gives you an idea of how much of the variability of the dependent variable is explained by the independent variable under the estimated model.
The value of R²= 0.11 or 11% means that only 11% of the variability of the number of car deaths is due to the seat belt law.
It looks like the variable "seat belt law" isn't a good regressor.
d. What possible independent/predictor variables could you add to this model?
X: "increasing of traffic controls"
X: "decreasing the speed limits"
X: "opening road safety courses in the communities"
I hope it helps!
Answers
The solution is x = 9 and y = 4, meaning Brody would work 9 hours babysitting and 4 hours cleaning tables to satisfy both conditions (total hours ≤ 13 and total earnings ≥ $150).
Given:
Brody can work a maximum of 13 hours: x + y ≤ 13
Brody must earn at least $150: 10x + 15y ≥ 150
These are the two inequalities we need to solve graphically.
Graph the first inequality: x + y ≤ 13
This inequality represents the total number of hours Brody can work, which cannot exceed 13 hours. We'll plot the line x + y = 13 and shade the region below it.
Graph the second inequality: 10x + 15y ≥ 150
This inequality represents the total earnings Brody needs to make, which should be at least $150. Let's simplify it to 2x + 3y ≥ 30. We'll plot the line 2x + 3y = 30 and shade the region above it.
Now, let's find the point where the shaded regions of both inequalities overlap. This point will represent the feasible solution where Brody's working hours and earnings satisfy both conditions.
Solving the system of inequalities graphically, you will find the point of intersection. However, since I can't create a graphical representation here, I'll explain how to calculate the solution point algebraically:
First, solve the equation x + y = 13 for y:
y = 13 - x
Now substitute this value of y into the equation 2x + 3y = 30:
2x + 3(13 - x) = 30
2x + 39 - 3x = 30
-x = -9
x = 9
Now substitute the value of x back into the equation y = 13 - x:
y = 13 - 9
y = 4
So, the solution is x = 9 and y = 4, meaning Brody would work 9 hours babysitting and 4 hours cleaning tables to satisfy both conditions (total hours ≤ 13 and total earnings ≥ $150).
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Graph y≤13−x (shading down)
graph y≥10− 3/2x (shading up)