Chapter 14.4, Problem 27E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

** Manufacturing** Find the values for each of the dimensions of an open-top box of length

(in inches) such that the box requires the least amount of material to make.

To determine

**To calculate:** The values of each of the dimensions of an open-top box of length *x*, width *y*, and height

Explanation

**Given Information:**

The dimensions of an open-top box are given by length *x*, width *y*, and height

**Formula used:**

To calculate relative maxima and minima of the function

(1) Find the partial derivatives

(2) Find the critical points, that is, the point(s) that satisfy

(3) Then find all the second partial derivatives and evaluate the value of *D* at each critical point, where

(a) If

(b) If

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

For a function

(1) When both derivatives are taken with respect to *x* is

(2) When both derivatives are taken with respect to *y* is

(3) When first derivative is taken with respect to *x* and second derivative is taken with respect to *y* is

(4) When first derivative is taken with respect to *y* and second derivative is taken with respect to *x* is

Power of

Quotient rule for function

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

**Calculation:**

Consider the problem, the dimensions of an open-top box are given by length *x*, width *y*, and height

Use the formula for the surface area of a box with open top,

Substitute

Thus, the area function is

Use the power of *x* rule for derivatives, the constant function rule, the quotient rule, and the coefficient rule,

Thus,

Differentiate *x* by holding *y* constant,

Equate the first derivative of function *x* to zero.

The first partial derivative of function *y* by holding *x* constant.

Equate the first derivative of function *y* to zero.

Now, calculate value of *x* and *y*

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