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3 years ago

What are the dimensions of the lightest open-top right circular cylindrical can that will hold a volume of 2197 cm cubed?

Mathematics

1 answer:

Zarrin [17]3 years ago

4 0

Answer:

radius = height ≈ 8.88 cm

Step-by-step explanation:

If the weight is proportional to the surface area, then we need to find the minimum surface area for the given volume.

The volume of the can is given by ...

V = πr²h . . . . . . for radius r and height h

The surface area includes that of the bottom and the sides:

SA = πr² +2πrh

Solving the first equation for h, we can write the second equation as ...

SA = πr² +2πr(V/(πr²)) = πr² +2V/r

The minimum area for the volume will be found where the derivative is zero.

d(SA)/dr = 2πr -2V/r² = 0

πr³ = V . . . . . multiply by r²/2 and add V

Then the can radius is ...

r = ∛(V/π)

and the height is ...

h = V/(π(V/π)^(2/3)) = ∛(V/π) = r

For the given volume, the dimensions are ...

r = h = ∛(2197/π) ≈ 8.8762 cm

The lightest can with that volume has a radius and height of about 8.88 cm.

_____

<em>Comment on cylinder aspect ratio</em>

You may notice that the lateral area is equal to twice the bottom area for this cylinder. If the can had a top, the minimum weight would be such that the lateral area is twice the total of top and bottom area. In that case, the height is equal to the diameter, not the radius.

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The alternate hypothesis is $H_a: \mu < 14$

The test statistic is t = -1.95.

The p-value is of 0.0292. This means that for a level of significance of 0.0292 and higher, there is sufficient evidence to conclude that the highlighters wrote for less than 14 continuous hours.

Step-by-step explanation:

Suppose a consumer product researcher wanted to find out whether a highlighter lasted less than the manufacturer's claim that their highlighters could write continuously for 14 hours.

At the null hypothesis, we test if the mean is 14 hours, that is:

$H_0: \mu = 14$

At the alternate hypothesis, we test if the mean is less than 14 hours, that is:

$H_a: \mu < 14$

The test statistic is:

$t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}$

In which X is the sample mean, $\mu$ is the value tested at the null hypothesis, s is the standard deviation of the sample and n is the size of the sample.

14 is tested at the null hypothesis:

This means that $\mu = 14$

X = 13.6 hours, s = 1.3 hours. Sample of 40:

In addition to the values of X and s given, we have that $n = 40$

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$t = -1.95$

The test statistic is t = -1.95.

P-value:

The p-value of the test is the probability of finding a sample mean lower than 13.6, which is a left tailed test, with t = -1.95 and 40 - 1 = 39 degrees of freedom.

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Answer:

Probability of stopping the machine when $X < 9$ is 0.0002

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Step-by-step explanation:

There is a random binomial variable $X$ that represents the number of units come off the line within product specifications in a review of $n$ Bernoulli-type trials with probability of success $0.91$ . Therefore, the model is ${15 \choose x} (0.91) ^ {x} (0.09) ^ {(15-x)}$ . So:

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What are the dimensions of the lightest​ open-top right circular cylindrical can that will hold a volume of 2197 cm cubed​?

What are the dimensions of the lightest open-top right circular cylindrical can that will hold a volume of 2197 cm cubed?