Question

You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs $7/ft2 and the material for the sides costs $3/ft2. You need a box with volume 4 ft3. Find the dimensions (in ft) of the box that minimize cost. Use x to represent the length of the side of the box and h to represent the height. (Round your answers to two decimal places.)

Answer 1

Answer:

The required dimension is 1.59 ft

Step-by-step explanation:

Volume of the box is V=XxHxW

Where

X = Length

H = height

W = width

Assume the box is cube. So, X=H=W=Y

V=Y3

Substituting the value of volume

4 = Y3

Taking cube root on both sides

(4)^(1/3) = Y^(3/3)

1.59 = Y

So the required dimensions are

X = H = W = 1.59 ft

Answer 2

Answer: height (h) = 1.59ft

length (x) = 1.59ft

width = 1.59ft

Total cost is $25.28 at these dimensions

Step-by-step explanation:

The volume of a box is given by multiplying its length by its width and height.

If x = length

h = height

it's safe to assume the box is a cube (having a square bottom): length = height = width

Volume = length x width x height = x³

Volume = x³

4 ft² = x³

x = 4^(1/3)

x = 1.587 =1.59ft approximately

Area of square bottom = 1.59 x 1.59 = 2.528ft²

Cost of square bottom = 2.528 x $7 (per ft²) = $17.698

Area of side = 2.528ft²

Cost of side = 2.528 x $3 (per ft²) = $7.584

Total cost = $17.698 + $7.584 = $25.28 to two decimal places.