A certain rectangular crate measures 8 feet by 10 feet by 12

Volume of the cylinder equals to \(area=\pi{r^2}h\). First of all note that answer choices C, D, and E don't make sense. For example cylinder of a radius 6 (option C) just won't fit on any face, as max face has a dimensions 12*10 so cylinder with max radius of 5 can be placed on it.

Max volume will be when the base of a cylinder is placed on the face with dimension 12*10 thus the radius will be 5 --> \(v=\pi{5^2}*8=200\pi\);

Other options:
If we place cylinder on the face with dimension 12*8 then radius will be 4 and \(v=\pi{4^2}*10=160\pi\);
If we place cylinder on the face with dimension 10*8 then radius will be 4 and \(v=\pi{4^2}*12=192\pi\),.

Answer: B.
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The dimensions of the crate are 8*10*12. The rectangular base of the crate can be 8*10 or 10*12 or 8*12 and the height will be the leftover dimension.
To maximize volume of the cylinder, we must maximize radius and height. The diameter of the cylinder will be the measure of the shorter side of the base i.e. if the base of the crate measures 8*10, the diameter will be 8. So one thing is clear - the two sides of the crate should be as close as possible to each other in measure because some space gets wasted. Hence having the base as 8*12 and height as 10 doesn't make sense. It is much better to have base as 8*10 and height as 12 since diameter will be 8 in both cases but height will be 12 in the second case.

Now, radius gets squared so larger the radius, more impact it will have on volume as compared with height. But radius is half of diameter so some impact is lost. Let's review both leftover cases:

Base 8*10, height 12
Volume of cylinder \(= \pi*4^2*12 = 192*\pi\)
.
Base 10*12, height 8
Volume of cylinder \(= \pi*5^2*8 = 200*\pi\)

The volume will be maximum when base is 10*12 (so radius is 5).

Answer (B)
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Good explaination

I don't agree with the solution, the radius can be 6. Also if you consider it will less than 6, it will be more of about 5.9 which is closer to 6 than 5. So i will mark 6 in this case.

The radius cannot be 6. It won't fit on any face.
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12x10. its close enough to have the radius as 6.

The radius of 6 means that the diameter is 12. So, the base of the cylinder will be 12 by 12 and it won't fit.
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yaa. I see .Good question. Thanks!

Make the base of the crate as 10 feet by 12 feet
The biggest object that can fit in this base is 10 feet by 10 feet; If we take an object with dimension of 11 feet then it will pop 1 feet out out that side of the base that is 10 feet.
Therefore the biggest cylinder has to have a diameter of 10 and not more than that
Radius will be half of diameter= 10/2 = 5
5 feet is the answer
So the corect option is B

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Can someone explain how we can conclude the radius of 6 won't fit on a 12 x 10 face? I am not sure I am following what is being said.

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