A closed cylindrical can has surface area 120 Pi square inches. Express the volu

Question

A closed cylindrical can has surface area 120 Pi square inches. Express the volume of the can as a function of it's radius. You are given that a cylinder of radius "r" and height "h" has a volume V= pi r squared h and lateral (side) surface area S = 2 pi r h. Also, a disk of radius "r" has area A = pi r squared. Here is
what I did: 1) I solved the area equation for "h". Here is my equation 2 pi r h 2 pi r squared
= 120 pi. With "h" isolated I got h = 120 pi - 2 pi r squared in the numerator and 2 pi r in the denominator.
Simplifying, I got h = (60-r). When I plug this solution for "h" back into the Volume equation, I get
V = pi r squared ( 60 - r). The book gets a different answer. The book has V = pi r ( 60 - r squared)
. Did I do something wrong or is the book answer wrong? I can't seem to arrive at the books answer, any insight would be tremendously helpful. Thanks!

Explanation / Answer

surface area = lateral surface area + upper disk area + lower disk area

= 2rh + r2 + r2 = 120

Thus:

2rh + 2r2 = 120 --> rh + r2 = 60

Thus:

h = (60 - r2)/r

So the volume is:

V = r2h = r2 (60 - r2)/r = r (60 - r2)

So the volume as a function of r is:

V(r) = r (60 - r2)

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