SOLUTION: A cone-shaped paper cup is to hold 100 cubic cm of water. Find the height and the radius of the cup that can be made from the least amout of paper.

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Question 133749: A cone-shaped paper cup is to hold 100 cubic cm of water. Find the height and the radius of the cup that can be made from the least amout of paper.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A cone-shaped paper cup is to hold 100 cubic cm of water. Find the height and the radius of the cup that can be made from the least amount of paper.
:
Use the volume of a cone formula: 1%2F3*pi*r^2*h = V; to find h in terms of r.
1%2F3*pi*r^2*h = 100
multiply equation 3 to get rid of the denominator
pi*r^2*h = 300
h = 300%2F%28%28pi%2Ar%5E2%29%29
:
Using the surface area equation: SA = pi*r^2 + (pi*r*L), find L using r and h
L = sqrt%28h%5E2+%2B+r%5E2%29
Substitute 300%2F%28%28pi%2Ar%5E2%29%29 for h
L = sqrt%28%28300%2F%28%28pi%2Ar%5E2%29%29%29+%2B+r%5E2%29
Substitute above for L in the SA equation
:
SA = (pi%2Ar%5E2) + (pi%2Ar*sqrt%28%28300%2F%28%28pi%2Ar%5E2%29%29%29+%2B+r%5E2%29)
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Using this equation in my TI83, the graph showed the minimum radius to occur at appox 2.37 cm
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Find the height using this value:
h = 300%2F%28%28pi%2A2.37%5E2%29%29
h = 17 cm
;
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Check solution by finding the volume
V = 1%2F3 * pi * 2.37^2 * 17
V = 99.994 ~ 100 cm
:
I was unable to produce the graph here. Hope this helps you.