   Chapter 8.5, Problem 45E

Chapter
Section
Textbook Problem

# Modeling Data The predicted cost C (in hundreds of thousands of dollars) for a company to remove p% of a chemical from its waste water is shown in the table. P 0 10 20 30 40 C 0 0.7 1.0 1.3 1.7 P 50 60 70 80 90 C 2.0 2.7 3.6 5.5 11.2 A model fat the data is given by C = 124 p ( 10 + p ) ( 100 − p ) for 0 ≤ p < 100 . Use the model to find the average cost of removing between 75% and 80% of the chemical.

To determine

To calculate: The average cost of removal of 75% to 80% of the chemical from its waste water.

Explanation

Given: The following table representing the relation between P and C;

And a model for the data is given by the following expression;

y=124p(10+p)(100p)

Formula used:

1. [F(x)]ab=F(b)F(a)

2. [f(u)±g(u)]du=f(u)du±g(u)du

3. 1xdx=lnx+C

Calculation:

Consider the following function.

y=124p(10+p)(100p)

Average cost of removing p% of chemical from waste water is,

180757580124p(10+p)(100p)dp

Then,

124p(10+p)(100p)dp=124p(10+p)(100p)dp

By using partial fraction method, we get,

p(10+p)(100p)=A10+p+B100pp(10+p)(100p)=A(100p)+B(10+p)(10+p)(100p)

By simplifying further, we get,

p=A(100p)+B(10+p)

Now solve for A, substitute p=10.

A(100+10)+B(1010)=10110A=10A=111

Then solve for B, substitute p=100.

A(100100)+B(10+100)=100110B=100B=1011

So, by substituting the value of A and B we get,

p(10+p)(100p)=11110+p+1011100p=111(10+p)+1011(100p)=111(p+10)1011(p100)......

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