Question

A storage box with a square base must have a volume of 80 cubic centimeters. The top and bottom cost \(\$ 0.20\) per square centimeter and the sides cost \(\$ 0.10\) per square centimeter. Find the dimensions that will minimize cost.

Short answer

The dimensions that will minimize the cost for a box with a volume of 80 cubic centimeters are \( x = \sqrt[3]{40} \) cm width and length, and \( h = \frac{80}{( \sqrt[3]{40})^2} \) cm height.

Step-by-step solution

Establish relationships

According to the problem, the box is square and has a volume of 80 cubic centimeters. So if we let x denote the length of the side and h denotes the height of the box. We then have the volume \( V = x^2h \). Given \( V = 80 \), we can express h in terms of x, i.e., \( h = \frac{80}{x^2} \). Additionally, the cost C can be expressed as \( C = 0.20*(2*x^2) + 0.10*(4*x*h) \), taking into account the manufacturing cost for the top, bottom, and sides of the box.

Form cost function

Replacing h in the cost function with \( h = \frac{80}{x^2} \), we get a function with only one variable - \( C = 0.20*(2*x^2) + 0.10*(4*x*(\frac{80}{x^2})) \). This can further be simplified to \( C = 0.4*x^2 + 32/x \).

Compute the derivative and solve for x

To find the minimum cost, we differentiate the cost function with respect to c and set it equal to zero, and solve for x, i.e., \( C' = 0.8*x - 32/(x^2) = 0 \). Solving this equation for x yields \( x = \sqrt[3]{40} \).

Verify that it is indeed an minimum

Verify that \( x = \sqrt[3]{40} \) does indeed represent a minimum cost, find the second derivative of the function and check whether it's positive at the calculated x value.

Calculate the dimensions of the box

Finally, compute the box’s dimensions. Having found \( x = \sqrt[3]{40} \), you can find h using \( h = \frac{80}{x^2} \).