Cylinder in a rectangle

[rahulvsd]

A large rectangular box is 4 feet wide, 5 feet long, and 8 feet tall. A metal cylinder is to be fitted snugly into the box and stood upright on its base, resting on any one of the 6 sides of the box. Of all such cylinders that could fit into the box, what is the diameter, in feet, of the one that has the largest possible volume?

a. 2 b. 2.5 C. 4 d. 5 e. 8

[spoiler]OA: C[/spoiler]

[Neo Anderson]

Refer the attached figure:-

there are three options to fit the cylinder in the rectangle
the volume of the cylinder is (Pie r^2 h) or (pie d^2 h/4)
thus for three options, the volume:-

option 1 d=5; h=4 => d^2 X h=> 100

option 2 d=4; h=5 => d^2 X h=> 80

option 3 d=4; h=8 => d^2 X h=> 128

clearly the volume with diameter of 4 will be maximum, hence c.

[Mike@Magoosh]

Hi, there. I'm happy to help with this.

I've attached a pdf with the full solution.

Let me know if anyone reading this has any questions.

Mike

[krusta80]

A large rectangular box is 4 feet wide, 5 feet long, and 8 feet tall. A metal cylinder is to be fitted snugly into the box and stood upright on its base, resting on any one of the 6 sides of the box. Of all such cylinders that could fit into the box, what is the diameter, in feet, of the one that has the largest possible volume?

a. 2 b. 2.5 C. 4 d. 5 e. 8

[spoiler]OA: C[/spoiler]

Without getting into any big-time calculations, let's look at it this way...

There are three possible surface sizes on which to rest the cylinder, and the max diameter allowed for each surface is the minimum of the two dimensions of that surface. Therefore, there are only two possible diameter lengths: 4 feet and 5 feet. This eliminates all but choices C and D.

The maximum height possible for a five-foot diameter is 4 feet, because 8 feet has to be paired with 5 feet to allow for a 5-foot diameter. The maximum height possible for a four-foot diameter is 8 feet. Now we need to calculate:

D = 5 --> Volume = 25*pi*4 = 100*pi
D = 4 --> Volume = 16*pi*8 = 128*pi

The answer is C