A cone-shaped paper drinking cup is to be made to hold 30 cm^3 of water. Find the height and radius of the cup that will use the smallest amount of paper. (Round your answers to two decimal places.)

Height = ?

1. 👍 0
2. 👎 0
3. 👁 1,661
1. 1/3 pi r^2 h = 30, so
h = 90/(pi r^2)

surface area = 2 pi r s
where r^2+h^2 = s^2, so

a = 2 pi r √(r^2 + (90/(pi r^2))^2)
= 2/r √(pi^2 r^6 + 8100)

for minimum paper, we need da/dr = 0, so, as wolframalpha so ably shows at

http://www.wolframalpha.com/input/?i=2+pi+r+%E2%88%9A%28r^2+%2B+%2890%2F%28pi+r^2%29%29^2%29

r = ∛(45√2/pi)
so, h = ∛(180/pi)

1. 👍 1
2. 👎 4

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