Answer:
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
Step-by-step explanation: