# Largest box that can be made from rectangular cardboard

Date of Exam

Subject

**Problem**

Mar wants to make a box with no lid from a rectangular sheet of cardboard that is 18 inches by 24 inches. The box is to be made by cutting a square of side *x* from each corner of the sheet and folding up the sides. Find the value of *x* that maximizes the volume of the box.

A. 4.3 inches | C. 10.6 inches |

B. 5.2 inches | D. 3.4 inches |

**Answer Key**

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[ D ]

**Solution**

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$V = (24 - 2x)(18 - 2x)x$

$V = 2x(12 - x)(9 - x)$

$V = 2x(108 - 21x + x^2)$

$V = 216x - 42x^2 + 2x^3$

To maximize, differentiate and equate to zero

$\dfrac{dV}{dx} = 216 - 84x + 6x^2 = 0$

$x^2 - 14x + 36 = 0$

$x = 10.6 \text{ and } 3.4$

$\text{Use } x = 3.4 \text{ inches}$ ← *answer*

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