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Problem Solving | Bold part focus

by [email protected] » Thu Jun 20, 2013 11:56 am
bThe inside dimensions of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindricalcanister 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 maximum volume?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

I am not able to understand why cant we take 10 as the length so that radius becomes 5 and height is 8. giving us the maximum volume?
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by Brent@GMATPrepNow » Thu Jun 20, 2013 12:01 pm
[email protected] wrote:bThe inside dimensions of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindricalcanister 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 maximum volume?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

I am not able to understand why cant we take 10 as the length so that radius becomes 5 and height is 8. giving us the maximum volume?
Volume of cylinder = pi(radius^2)(height)

There are 3 different ways to position the cylinder (with the base on a different side each time).
You can place the base on the 6x8 side, on the 6x10 side, or on the 8x10 side

If you place the base on the 6x8 side, then the cylinder will have height 10, and the maximum radius of the cylinder will be 3 (i.e., diameter of 6).
So, the volume of this cylinder will be (pi)(3^2)(10), which equals 90(pi)

If you place the base on the 6X10 side, then the cylinder will have height 8, and the maximum radius of the cylinder will be 3 (i.e., diameter of 6).
So, the volume of this cylinder will be (pi)(3^2)(8), which equals 72(pi)

If you place the base on the 8x10 side, then the cylinder will have height 6, and the maximum radius of the cylinder will be 4 (i.e., diameter of 8).
So, the volume of this cylinder will be (pi)(4^2)(6), which equals 96(pi)

So, the greatest possible volume is 96(pi) and this occurs when the radius is 4

Cheers,
Brent
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by faraz_jeddah » Fri Jun 21, 2013 6:22 am
Brent@GMATPrepNow wrote:
[email protected] wrote:bThe inside dimensions of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindricalcanister 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 maximum volume?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

I am not able to understand why cant we take 10 as the length so that radius becomes 5 and height is 8. giving us the maximum volume?
Volume of cylinder = pi(radius^2)(height)

There are 3 different ways to position the cylinder (with the base on a different side each time).
You can place the base on the 6x8 side, on the 6x10 side, or on the 8x10 side

If you place the base on the 6x8 side, then the cylinder will have height 10, and the maximum radius of the cylinder will be 3 (i.e., diameter of 6).
So, the volume of this cylinder will be (pi)(3^2)(10), which equals 90(pi)

If you place the base on the 6X10 side, then the cylinder will have height 8, and the maximum radius of the cylinder will be 3 (i.e., diameter of 6).
So, the volume of this cylinder will be (pi)(3^2)(8), which equals 72(pi)

If you place the base on the 8x10 side, then the cylinder will have height 6, and the maximum radius of the cylinder will be 4 (i.e., diameter of 8).
So, the volume of this cylinder will be (pi)(4^2)(6), which equals 96(pi)

So, the greatest possible volume is 96(pi) and this occurs when the radius is 4

Cheers,
Brent
Hi Brent - you didnt answer the question that was asked - "I am not able to understand why cant we take 10 as the length so that radius becomes 5 and height is 8. giving us the maximum volume?"
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by Brent@GMATPrepNow » Fri Jun 21, 2013 6:28 am
faraz_jeddah wrote: Hi Brent - you didnt answer the question that was asked - "I am not able to understand why cant we take 10 as the length so that radius becomes 5 and height is 8. giving us the maximum volume?"
Sorry about that - I wasn't sure what you meant by "taking 10 as the length," so I just showed the entire solution.

If you want the height of the cylinder to be 8, then the flat, circular part of the cylinder must rest on the side of the box with dimensions 6 by 10.
A circle on a 6 by 10 rectangle cannot have a radius of 5, since the diameter would then be 10 inches, which is too big (since the rectangle is only 6 inches wide)

Cheers,
Brent
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