   Chapter 9.7, Problem 41E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Volume When squares of side x inches are cut from the corners of a 12-inch-square piece of cardboard, an open-top box can be formed by folding up the sides. The volume of this box is given by V = x ( 12 − 2 x ) 2 Find the rate of change of volume with respect to the size of the squares.

To determine

To calculate: The rate of change of volume of the box with the volume function, V=x(122x)2.

Explanation

Given Information:

A square of side 12-inch is cut from the corners in the shape of squares with side x inches. The volume of the box is represented by the function, V=x(122x)2.

Formula used:

According to the power rule, if f(x)=xn, then,

f(x)=nxn1

According to the property of differentiation, if a function is of the form, g(x)=cf(x), then,

g(x)=cf(x)

According to the property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

f(x)=u(x)+v(x)

According to the product rule, if f(x)=u(x)v(x), then

f(x)=u(x)v(x)+v(x)u(x)

The derivative of a constant value, k, is

ddx(k)=0

According to the property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nun1dudx

Calculation:

Consider the provided volume function,

V=x(122x)2

The rate of change of volume is found by differentiating the volume function of the box with respect to x.

Consider (122x) to be u,

V=xu2

Differentiate both sides with respect to x,

V=ddx(xu2)

Simplify using the product rule,

V=(ddx(x))u2+(ddx(u2))x

Simplify using the power rule,

V=(x11)u2+(2u21dudx)x=

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