SOLUTION: The volume of rectangular solid with a square base is 216 cubic inches.What is the least possible surface area for the solid? (A) 24 in.2 (B) 72 in.2 (C) 216 in.2 (D) 252 in.2

Algebra ->  Customizable Word Problem Solvers  -> Geometry -> SOLUTION: The volume of rectangular solid with a square base is 216 cubic inches.What is the least possible surface area for the solid? (A) 24 in.2 (B) 72 in.2 (C) 216 in.2 (D) 252 in.2       Log On

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Question 332495: The volume of rectangular solid with a square base is 216 cubic inches.What is
the least possible surface area for the solid?
(A) 24 in.2 (B) 72 in.2 (C) 216 in.2 (D) 252 in.2 (E) 492 in.2
Answer by jrfrunner(365) About Me  (Show Source):
You can put this solution on YOUR website!
This requires knowing the equation of the volume and surface area of rectangles
volume: (area of base)*height,
since base is a square and if its side is "S" then
volume+=+s%5E2%2AH
Surface area: 2* (area of the base) + (perimeter of the base) * height
Surface+Area+=2%2AS%5E2+%2B+4%2AS%2AH
====
Solve the volume equation for H: H=Volume%2F%28S%5E2%29
substitute this into the surface area: 2%2AS%5E2%2B4%2AS%2AH+=+2%2AS%5E2+%2B+4%2AS%28Volume%2F%28S%5E2%29%29
Surface area= SA=2%2AS%5E2+%2B+4%2Avolume%2FS+=+2%2AS%5E2+%2B+4%2A%28216%29%2FS+=+2%2AS%5E2%2B864%2FS
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Take the derivative of SA: (SA)' =+4%2AS+-864%2F%28S%5E2%29
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set this derivative =0 and solve for S
4S-864%2F%28S%5E2%29+=+0
4S=864%2F%28S%5E2%29
4S%5E3=864
S%5E3=216
S=6
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Check to see if this a minimum or maximum by taking the second derivative of SA
(SA)" =+4-864%2A%28-2%29%2F%28S%5E3%29=4%2B3456%2F%28S%5E3%29 evaluated at S=6
(SA)"= 4%2B3456%2F%286%5E3%29=20, since its positive this means, the function curves upward, thus S=6 is a minimum
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Now substitute S=6 into the SA equation to find the lowest surface area
SA=+2%2AS%5E2%2B864%2FS+=+2%2A6%5E2%2B864%2F6=2%2A36%2B144=216