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

To minimize the cost of constructing the cat box, we need to determine the dimensions that minimize the use of material given the cost of $7/ft² for the bottom and $3/ft² for the sides and a volume of 4 ft³. By expressing the height h in terms of x, we can then formulate a function to optimize using calculus techniques.

Step-by-step explanation:

To minimize the cost of constructing a box with a volume of 4 ft³, we need to find the dimensions that would result in the least amount of material used.

The cost of the material for the square bottom is $7/ft², while the material for the sides is $3/ft². We represent the length of the side of the square bottom by x and the height of the box by h. Since the bottom is a square, its area is x² and the volume of the box is x²h=4 ft³. We can express h in terms of x as h = 4/x².