Question 1121065
The volume of the rectangular prism is V = l*w*h = 84 in^3
The area is A = 2(l*w + l*h + w*h)
The base l is equal to twice the width: l = 2w
Thus V = 2*w^2*h and A = 2(w^2 + 2wh + wh) = 2w(w + 3h)
Expressing h in terms of w gives h = V/(2*w^2)
Thus A = 2w(w + 3V/(2*w^2))
The area will be minimized when dA/dw = 0
0 = 4w - 3V/w^2 -> w = (3V/4)^(1/3)
Substituting the value for V gives w = 3.979
Therefore l = 7.958 and h = 84/(2*3.979^2) = 2.653