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20 Dec 2015, 09:21
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Difficulty:
55%
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Question Stats:
60% (02:05) correct
40%
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wrong
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A crate measures 4 feet by 8 feet by 12 feet on the inside. A stone pillar in the shape of a right circular cylinder must fit into the crate for shipping so that it rests upright when the crate sits on at least one of its six sides. What is the radius, in feet, of the pillar with the largest volume that could still fit in the crate?
A. 2
B. 4
C. 6
D. 8
E. 12
A. 2
B. 4
C. 6
D. 8
E. 12
20 Dec 2015, 22:06
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A crate that is 4 by 8 by 12 has 3 faces.
Face 1: L=4, B=8, H=12. In trying to fit a circle into a 4 by 8 rectangle, diameter can only be maximized to be 4. As such, the volume of the cylinder is
pi X (4/2)^2 X 12 = 48pi
Face 2: L=4, B=12, H=8. Same as above, the circle has to fit into a 4 by 12 rectangle, diameter is 4. Volume of cylinder is
pi X (4/2)^2 X 8 = 32pi
Face 3: L=8, B=12, H=4. Lastly, the circle has to fit into a 8 by 12 rectangle, diameter is 8. Volume of cylinder is
pi X (8/2)^2 X 4 = 64pi
The volume in the 3rd scenario is the largest. As such, to maximize the volume of the cylinder, the radius has to be 8/2=4.
Face 1: L=4, B=8, H=12. In trying to fit a circle into a 4 by 8 rectangle, diameter can only be maximized to be 4. As such, the volume of the cylinder is
pi X (4/2)^2 X 12 = 48pi
Face 2: L=4, B=12, H=8. Same as above, the circle has to fit into a 4 by 12 rectangle, diameter is 4. Volume of cylinder is
pi X (4/2)^2 X 8 = 32pi
Face 3: L=8, B=12, H=4. Lastly, the circle has to fit into a 8 by 12 rectangle, diameter is 8. Volume of cylinder is
pi X (8/2)^2 X 4 = 64pi
The volume in the 3rd scenario is the largest. As such, to maximize the volume of the cylinder, the radius has to be 8/2=4.
20 Dec 2015, 22:33
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Sash143
We can find the radius of all the three cases of cylinders.
The only crux to find the answer faster is that:
Voulme is pi*r^2*h. The volume is a function of r^2. so r has to be the highest to find the largest volume.
So r=4 for the surface 8*12 face.
Volume = 64pi
