Question

An open box of maximum volume is to be made from a square piece of material, 24 inches on a side, by cutting equal squares from the corners and turning up the sides (see figure). (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maximum volume. $$ \begin{array}{|c|c|c|} \hline \text { Height } & \begin{array}{c} \text { Length and } \\ \text { Width } \end{array} & \text { Volume } \\ \hline 1 & 24-2(1) & 1[24-2(1)]^{2}=484 \\ \hline 2 & 24-2(2) & 2[24-2(2)]^{2}=800 \\ \hline \end{array} $$ (b) Write the volume \(V\) as a function of \(x\). (c) Use calculus to find the critical number of the function in part (b) and find the maximum value. (d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.

Short answer

The maximum volume of the box is obtained at the critical number found by setting the derivative of the volume function \(f(x)=x(24-2x)^2\) equal to zero.

Step-by-step solution

Complete the Table

Proceed to complete the table for the next four heights, that is, x=3, x=4, x=5, and x=6. For every row, calculate the Length and Width by using \(24-2x\) and the volume by using \(x[24-2x]^2\).

Formulate the Volume function

Formulate the volume function as a function of \(x\). The volume \(V\) is given by length * width * height. Since the length and width are the same, the volume function \(V\) is given by \(V=x(24-2x)^2\).

Find the Critical Number

Utilize calculus to find the critical number for the function defined in the prior step. The critical number of a function is where the derivative equals zero or is undefined. Derive \(V=x(24-2x)^2\), set it equal to 0, and solve for \(x\). That gives the critical number.

Use Calculus to Find the Maximum Value

To find the maximum value, use the second derivative test. Calculate the second derivative of the volume function and substitute the critical number. If the result is less than zero, the volume function has a maximum at the critical number. Calculate the maximum volume by substituting the critical number in the volume function.

Graph the Volume Function

Utilize a graphing tool to represent the function \(f(x)=x(24-2x)^2\) and confirm the maximum volume visually. This graph will peak at the critical number and the height of the graph at this point is the maximum volume.