Question

The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs nine times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. (Let x, y, and z be the dimensions of the aquarium. Enter your answer in terms of V.) (x, y, z)

Answer 1

To minimize the cost of materials, we need to consider the areas of both the base and the sides of the aquarium. The dimensions that minimize the cost of the materials are x = y = √(V/9) and z = V / (9 * √(V/9)).

Step-by-step explanation:

To minimize the cost of materials, we need to consider the areas of both the base and the sides of the aquarium.

Let's assume that the dimensions of the base are 'x' and 'y', and the height of the aquarium (the side made of glass) is 'z'.

The volume of the aquarium is given by V = x * y * z.

To minimize the cost, we need to minimize the total surface area of the aquarium. The cost of the slate base is 9 times that of the glass side, so we want to minimize the area of the base.

Since the area of the base is x * y, we can rewrite the volume equation as V = (9 * x * y) * z.

Thus, the dimensions that minimize the cost of the materials are x = y = √(V/9) and z = V / (9 * √(V/9)).