Chapter 14.4, Problem 28E

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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 a closed-top box of length

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

To determine

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

Explanation

**Given Information:**

The dimensions of a closed-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 closed top and length *l*, width *w*, and height *h* is

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

**Calculation:**

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

Use the formula for the surface area of a box with closed top and length *l*, width *w*, and height *h*.

Substitute

Use the formula for the volume of a box with length *l*, width *w*, and height *h*.

Substitute

Since, the volume of the box is

Substitute *V*.

Substitute

Thus, the area function is

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

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