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A rectangular box has the dimensions 12 inches x 10 inches x

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A rectangular box has the dimensions 12 inches x 10 inches x [#permalink]  14 Oct 2007, 16:45
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A rectangular box has the dimensions 12 inches x 10 inches x 8 inches. What is the largest possible volume of a right cylinder that is placed inside the box?

Answer: The radius of the cylinder must be equal to half of the smaller of the 2 dimensions that form the box's bottom. ............... and teh answer is 200 x pi

I just don't understand why does the radius half to be equal to half of the smaller of the 2 dimensions. does anyone understand why? thx.
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[#permalink]  14 Oct 2007, 18:05
well, if the base of the cilinder stands lets say on the side 10 by 8, to make it fit in the rectangular box, radius of cilinder has to be 4, it won't just fit any other way. And if it stands on 12 by 10 side, radius has to be 5. Just try drawing it...
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[#permalink]  14 Oct 2007, 20:40
There are 3 options here.

1) The height (H) of the cylinder is 12 so the Max diameter (D) it can have is 8. So the area (A)= 192

2) H = 8 and max. D = 10. So A = 200

3) H = 10 and max. D = 8. So A = 160.

Therefore the answer is 200 (pi)
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Re: geometry question [#permalink]  14 Oct 2007, 21:18
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jimjohn wrote:
A rectangular box has the dimensions 12 inches x 10 inches x 8 inches. What is the largest possible volume of a right cylinder that is placed inside the box?

Answer: The radius of the cylinder must be equal to half of the smaller of the 2 dimensions that form the box's bottom. ............... and teh answer is 200 x pi

I just don't understand why does the radius half to be equal to half of the smaller of the 2 dimensions. does anyone understand why? thx.

Its helpful to draw this out.

1st Cyl. L,W,H : 12,10,8
Greatest diameter we can have is 10. (can't be 12 b/c it wouldnt fit in the box). So radius is equal to 5.

pir^2h = Vol of Cyl.

pi5^2*8 25*8=200.

2nd Cyl: L,W,H: 10,8,12

pi4^2*12 --> 16*12 =192

3rd Cyl: L,W,H: 8,10,12

No need to calculate as we can see this is going to be smaller than 2.

Dimensions L,W,H: 12,10,8 result in the greatest volume for a cylinder.

In answer to ur question. Draw a square, forget the cube.

Do dimensions 12 and 10. Now draw a line from the midpoint of one side of the square to another side of the square.

now draw a circle. you will see that only when the diameter is 10 will it actually fit in the box.
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Re: geometry question [#permalink]  14 Oct 2007, 21:31
jimjohn wrote:
A rectangular box has the dimensions 12 inches x 10 inches x 8 inches. What is the largest possible volume of a right cylinder that is placed inside the box?

Answer: The radius of the cylinder must be equal to half of the smaller of the 2 dimensions that form the box's bottom. ............... and teh answer is 200 x pi

I just don't understand why does the radius half to be equal to half of the smaller of the 2 dimensions. does anyone understand why? thx.

the possible dimeters are either 10 or 8 but it cannot be 12. to have 12 as dimeter, we need another dimension at least of 12 or grater. so

1: if d = 10, then h = 8
so v = pi r^2 h = pi (5)^2 8 = 200 pi

2: if d = 8, h = 12
so v = pi r^2 h = pi (4)^2 12 = 192 pi

so first is the max.
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Re: geometry question [#permalink]  18 Oct 2007, 20:37
jimjohn wrote:
A rectangular box has the dimensions 12 inches x 10 inches x 8 inches. What is the largest possible volume of a right cylinder that is placed inside the box?

Answer: The radius of the cylinder must be equal to half of the smaller of the 2 dimensions that form the box's bottom. ............... and teh answer is 200 x pi

I just don't understand why does the radius half to be equal to half of the smaller of the 2 dimensions. does anyone understand why? thx.

200pi is max volume.

The height of cylinder can be 12, 10 or 8

if height is 12, max diameter is 8----------> Volume is 12 x 16pi=192pi
if height is 10, max diameter is 8----------> Volume is 10 x 16pi=160pi
if height is 8, max diameter is 10---------->Volume is 16 x 10pi=200pi
Re: geometry question   [#permalink] 18 Oct 2007, 20:37
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