solid geometry

Question

The inside dimensions of a rectangular wooden box are  6 inches by 8 inches by 10 inches .A cylindrical cansister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces .what is the possible volume  of all such cansisters that could be used?

 

(a)54 pi

(b)64 pi 

(c)72 pi
(d)90 pi

(e)96 pi

Answer 1

It sounds to me, based on the text "all such cansisters", that there are 3 answers; one for each "height" and base rectangle.

10" high with 6x8 base is one answer

8"   high with 10x6 base is another

6"   high with 8x10 base is the third.

 

Using the  formula for volume of a cylander V = pi r^2 * h , you can work out the answers as

1: pi 3^2 *10 == 90 pi

2: pi 3^2 * 8 == 72 pi 

3: pi 4^2 *6 ==  96 pi 

 

In the first 2 cases the radius/diameter is limited by the smallest side of the base rectangle, which is 6 (3" radius)

in case 3 we can use radius of 4"

 

I believe the question is worded such that a single container that will fit in any orientation is a valid interpretation,

 

Answer 2

Lets focus on the smallest side length of the box.  Suppose the height of the box is 6 inches.  Then the height of the canister can only be no more than 6 inches.  Any more, the canister will not fit inside the box.  The same goes for the radius.  This is because the canister will be sitting on top of the rectangular face that is 6 by 8 or 6 by 10, but inside the box. Since the radius will be half of the shortest side length, the radius will be 3.

 

In conclusion,

 

the radius is no more than 3 inches and the height is no more than 6 inches.

 

 

The formula for the volume of cylinders is

 

V = πr²h

 

 

One possible volume is

 

V = π(3²)(6)

 

V = 54π

 

 

54π is in our answer choices.

 

Your answer is (a).