SOLUTION: A cyndrical hot water heater holds 40 gallons. What height and radius minimizes surface area? 1 gallon= 231 inches cubed.

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Question 994359: A cyndrical hot water heater holds 40 gallons. What height and radius minimizes surface area? 1 gallon= 231 inches cubed.
Answer by htmentor(1343) About Me  (Show Source):
You can put this solution on YOUR website!
The surface area of a closed cylinder, A+=+2pi%2Ar%2Ah+%2B+2pi%2Ar%5E2
where r is the radius, and h is the height
Express h in terms of the volume, V which is known:
V+=+pi%2Ar%5E2%2Ah+-%3E+h+=+V%2F%28pi%2Ar%5E2%29
So we can express the area as:
A+=+2pi%2Ar%2AV%2F%28pi%2Ar%5E2%29+%2B+2pi%2Ar%5E2+-%3E+A+=+2V%2Fr+%2B+2pi%2Ar%5E2
The surface area is minimized when dA%2Fdr+=+0+=+-2V%2Fr%5E2+%2B+4pi%2Ar
Solving for r gives r+=+%28V%2F%282pi%29%29%5E%281%2F3%29
V = 40 gallons * 231 ci/gallon = 9240 ci (cubic inches)
Therefore, r+=+%289240%2F%282pi%29%29%5E%281%2F3%29+=+11.372+inches
h+=+9240%2F%28pi%2A11.372%5E2%29+=+22.744+inches