Numerical, Graphical, and Analytic Analysis 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.
(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.
Step 1 of 5) Exploration In Example 7, the rate of conversion was assumed to be proportional to the square of the unconverted amount. How would the result change if the rate of conversion were assumed to be proportional to the unconverted amount