Chapter 10.4, Problem 28E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

*Volume*

(a) A square piece of cardboard 36 centimeters on a side is to be formed into a rectangular box by cutting squares with length *x* from each corner and folding up the sides. What is the maximum volume possible for the box?

(b) Show that if the piece of cardboard is *k* centimeters on each side, cutting squares of size *k/6* and folding up the sides gives the maximum volume.

**(a)**

To determine

**To calculate:** The maximum volume of the box.

Explanation

**Given Information:**

**Formula used:**

To find the maximum value, calculate the relevant stationary value of an equation. Differentiate the function with respect to the independent variable and equate it to

If the second derivative of the provided function is less than zero, then substituting the value of the independent variable will give the maximum value of the equation.

The product rule is used when the function is a product of two differentiable terms. If

Volume of a rectangular box is given by:

Where, *l* is length, *b* is breadth and *h* is height of the box.

**Calculation:**

As given,

Thus, two lengths are cut from each corner, that is *x* is height of the box.

The volume of the box is

To find the stationary value for

Put

**(b)**

To determine

**To prove:** Cutting squares of side

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