Question

a rectangular storage container must have a volume of 1,728 cubic inches. the material costs $0.75 per square inch of surface area. what surface area will minimize the production cost and what will the total cost be? the surface area that will minimize the production cost is square inches. the total cost for the materials will be

Explanation:

Step1: Let the side length of the cube

Since for a rectangular container with a given volume, the cube has the minimum surface - area. Let the side length of the cube be \(x\). The volume formula of a cube is \(V = x^{3}\). Given \(V=1728\), then \(x^{3}=1728\), so \(x = \sqrt[3]{1728}=12\).

Step2: Calculate the surface area of the cube

The surface - area formula of a cube is \(S = 6x^{2}\). Substitute \(x = 12\) into the formula, we get \(S=6\times12^{2}=6\times144 = 864\).

Step3: Calculate the total cost

The cost per square inch is \(C_{0}=0.75\). The total cost \(C\) is given by \(C = 0.75\times S\). Substitute \(S = 864\) into the formula, we get \(C=0.75\times864=648\).

Answer:

The surface area that will minimize the production cost is \(864\) square inches. The total cost for the materials will be \(\$648\).