   Chapter 9.5, Problem 54E ### 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

# Emissions The table shows data for sulfur dioxide emissions from electricity generation (in millions of short tons) for selected years from 2000 and projected to 2035. These data can be modeled by the function E ( x ) =   ( 0.001 x   −   0.062 ) ( − 0.18 x 2 +   8.2 x   −   200 ) where x is the number of years past 2000.(a) Find the function that models the rate of change of these emissions.(b) Find and interpret E ' ( 20 ) . Year Short Tons (in millions) 2000 11.4 2005 10.2 2008 7.6 2015 4.7 2020 4.2 2025 3.8 2030 3.7 2035 3.8 Source: U.S. Department of Energy

(a)

To determine

To calculate: The function that models the rate of change of emissions for the sulfur dioxide, if the function of the emission of the sulfur dioxide from electricity generation (in millions of short tons) for selected years from 2000 and projected to 2035 modeled as, E(x)=(0.001x0.062)(0.18x2+8.2x200) where x is the number of years past 2000 and the table that shows the emissions of sulfur dioxide is shown below.

 Year Short Tons (in millions) 2000 11.4 2005 10.2 2008 7.6 2015 4.7 2020 4.2 2025 3.8 2030 3.7 2035 3.8
Explanation

Given Information:

The model of the emissions of sulfur dioxide is E(x)=(0.001x0.062)(0.18x2+8.2x200) and the table that gives the data for the emissions of theses sulfur dioxide with their respective years is tabulated below.

 Year Short Tons (in millions) 2000 11.4 2005 10.2 2008 7.6 2015 4.7 2020 4.2 2025 3.8 2030 3.7 2035 3.8

Formula Used:

The product rule for the derivative of the two function, ddx(fg)=fdgdx+gdfdx.

The sum and difference rule of derivate of functions, ddx[u(x)±v(x)]=ddxu(x)±ddxv(x).

The simple power rule of derivative ddx(xn)=nxn1.

Calculation:

Consider the provided function that models the rate of change of emissions for the sulfur dioxide, E(x)=(0.001x0.062)(0.18x2+8.2x200).

Differentiate the provided model function,

dE(x)dx=ddx[(0.001x0.062)(0.18x2+8.2x200)]

Use the product rule for the derivative of the two function f(x) and g(x) is, ddx(fg)=fdgdx+gdfdx.

dE(x)dx=(0.001x0.062)ddx(0.18x2+8.2x200)+(0

(b)

To determine

To calculate: The rate of change of the emission of sulfur dioxide E'(20) and its interpretation.

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