10th Edition

Ron Larson + 1 other

ISBN: 9781285057095

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.

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)(100−p)

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)(100−p)

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

180−75∫7580124p(10+p)(100−p)dp

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

By using partial fraction method, we get,

p(10+p)(100−p)=A10+p+B100−pp(10+p)(100−p)=A(100−p)+B(10+p)(10+p)(100−p)

By simplifying further, we get,

p=A(100−p)+B(10+p)

Now solve for A, substitute p=−10.

A(100+10)+B(10−10)=−10110A=−10A=−111

Then solve for B, substitute p=100.

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

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

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

Check out a sample textbook solution.

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