Suppose that you are to make a rectangular box with a square base from two different materials. The material for the top and four sides of the box costs $1/ft2$1/ft2; the material for the base costs $2/ft2$2/ft2. Find the dimensions of the box of greatest possible volume if you are allowed to spend $144 for the material to make it.

Answer 1

I will assume you are saying $1 per square foot

let the base be x ft by x ft, let the height be y
area of top and 4 sides = x^2 + 4xy
cost of top and 4 sides = 1(x^2 + 4x) = x^2 + 4xy

area of base = x^2
cost for the base = 2x^2
total cost = 3x^2 + 4xy = 114
y = (114 - 3x^2)/4x

V = x^2 y = x^2(114 - 3x^2)/(4x)
= (114/4)x - (3/4)x^3
dV/dx = 114/4 - (9/4)x^2 = 0 for a max of V
9/4 x^2 = 114/4
9x^2 = 114
x^2 = 114/9
x = √114/3 = appr 3.56 ft
y = appr 5.34 ft

check: V for my answer = appr 67.6215

let x = 3.5, then y = 5.5178
V = 67.59
let x = 3.6, then y = 5.217
V = 67.608

in both cases the volume was less than the volume for my dimension.
I have a very very strong feeling my answer is right.

Answer 2

the question asked for 144 not 114, if you plug in 144 you will get x=4