A rectangular box with a square base is open at the top and has a volume of 12 cubic feet. Each side of the base has a length of x feet. Which of the following expresses the surface area, S, in square feet, of the outside of the box in terms of x ? (CLEP precalculus test)
#1:S=x^2+(48/x) or #2: S= x^2+(24/X)I got #2 but the answer key said #1 is correct answer. It didn't explain Why? please show me how to solve it.

Since base = x, then area of base = x%5E2

Since volume of box = 12 cub. ft, and since Volume = LWH, then we can say that:

x(x)(H) = 12, with H being the height of the box

x%5E2H+=+12 x%5E2+=+12%2FH

x%5E2H+=+12 H+=+12%2Fx%5E2

Since S, or surface area of the OPEN box = L*W (base, or bottom) + 2*L*H (front & rear) + 2*W*H (left and right sides), then we have:

S+=+x%28x%29+%2B+2%28xH%29+%2B+2%28xH%29

S+=+x%5E2+%2B+2xH+%2B+2xH

S+=+x%5E2+%2B+4xH

S+=+x%5E2+%2B+4xH

S+=+x%5E2+%2B+4x%2812%2Fx%5E2%29 Substituting 12%2Fx%5E2 for H

S+=+x%5E2+%2B+4cross%28x%29%2812%2F%28x%29cross%28x%5E2%29%29 Canceling x in numerator and denominator

highlight_green%28S++=+x%5E2+%2B+48%2Fx%29

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