The inside dimensions of a rectangular wooden box
are 6 inches by 8 inches by 10 inches. A cylindrical
canister is to be placed inside the box so that it stands
upright when the closed box rests on one of its six
faces. Of all such canisters that could be used, what is
the radius, in inches, of the one that has the
maximum volume?
A-3
B-4
C-5
D-6
E-8

OA B

It is a bit difficult to explain without showing the figure. However, I will try my best.

For a box with dimensions 6 x 8 x 10, we can have 3 different rectangular faces.

6 x 8
8 x 10

and

6 x 10

For each of the faces, the maximum diameter of the cylinder that can fit its circular base and its corresponding height are as follows:

6 x 8 - Max diameter is 6 and Height is 10. Volume = PI x 3 x 3 x 10 = 90 x PI
6 x 10 - Max diameter is 6 and height is 8 Volume = PI x 3 x 3 x 8 = 72 x PI
8 x 10 - Max diameter is 8 and height is 6. Volume = PI x 4 x 4 x 6 = 96 x PI.

So the radius of the cylinder with max volume is 4.

Ans (B).

Hope this helps.

Thank you Harsih

There are three ways to place a cylindrical canister in the rectangular box:

1) Resting on the face measuring- 8 by 6
Volume = Pi(3^2)*10 = 90Pi
2) Resting on the face measuring - 8 by 10
Volume = Pi(4^2)*6 = 96Pi
3) Resting on the face measuring - 6 by 10
Volume = Pi(3^2)*8 = 72Pi

Volume is maximum in (2) when the radius is 4.

My question to you guys is why couldn't we have the the dimensions by 10x6, so 5=r and 8=h?

Thanks alot

The diameter cannot be 10 because the base of the cylinder is a circle and the circle base must fit inside the smallest dimension of the rectangular base.

If you have rectangular base of 10 and 8, the most the diameter of the circle (base of cylinder) can be is 8. Draw it out and try to fit a circle in that rectangle.

thanks very much for the explanation - absolutely makes sense

Nice work!

Rey