Chapter 14.5, Problem 26E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

** Manufacturing** Show that a box with a square base, an open top, and a fixed volume requires the least material to build if it has a height equal to one-half the length of one side of the base.

To determine

**To prove:** That a box has height equal to one half the length of one side of the base if the box has a square base, an open top, and fixed volume and required the least material to build.

Explanation

**Given Information:**

A box with a square base, an open top, and a fixed volume that requires the least material.

**Formula used:**

According to the Lagrange multipliers method to obtain maxima or minima for a function

Step 1: Find the critical values of

Step 2: The critical points of

Step 3: The critical points of

For a function *f* with respect to *y* is calculated by taking the derivative of *y* and keeping the other variable *x* constant. The partial derivative of *f* with respect to *y* is denoted by

Power of

Constant function rule for a constant

Coefficient rule for a constant

The surface area of a box with open top and length *l*, width *w*, and height *h* is

The volume of a box with length *l*, width *w*, and height *h* is

**Proof:**

Recall that, the surface area of a box with open top and length *l*, width *w*, and height *h* is

Let *x* be the length of the side of the square base and *y* be the height of the box.

Thus, the surface area of a box with open top and length *x*, width *x*, and height *h* is

Thus, minimize the function

Recall that, the volume of a box with length *l*, width *w*, and height *h* is

Since, *x* be the length of the side of the square base and *y* be the height of the box.

Thus, the volume of the box is given by

Let the fixed volume of the box be

Thus,

The provided constraint is

According to the Lagrange multipliers method,

The objective function is

Thus,

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