Q:

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.)

Accepted Solution

A:
Answer: height (h) = 1.59ftlength (x) = 1.59ftwidth = 1.59ftTotal 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 = heightit's safe to assume the box is a cube (having a square bottom): length = height = widthVolume = length x width x height = x³Volume = x³4 ft² = x³x = 4^(1/3)x = 1.587 =1.59ft approximately Area of square bottom = 1.59 x 1.59 = 2.528ft²Cost of square bottom = 2.528 x $7 (per ft²) = $17.698Area of side = 2.528ft²Cost of side = 2.528 x $3 (per ft²) = $7.584Total cost = $17.698 + $7.584 = $25.28 to two decimal places.