Polynomial Functions

Maya, a packaging engineer, designs an open-top fruit tray from a cardboard sheet measuring 40 by 30 centimeters. She cuts equal squares from each corner and folds up the sides to maximize the tray’s volume.

Suppose each cutout square has side length x. After folding, the tray's height is x, while its length and width each decrease by 2x. Multiplying these dimensions gives the volume function, which models how volume changes with cut size.

The cut size must be greater than 0 because the tray needs a positive height. It must also be less than 15 because a 15-centimeter cut from both sides reduces the width to zero. This interval is the practical domain in which the tray has positive dimensions.

Maya calculates the volumes for several cut sizes and plots the points. The graph rises, reaches its highest point, and then falls as the base becomes too small. She tests a few cut sizes near the peak to estimate the maximum volume.

The graph shows that a cut size of about 5.7 centimeters gives the maximum volume. Maya substitutes this value for x in the expressions for length, width, and height to find the tray’s dimensions.