   Chapter 6.1, Problem 7CP ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Checkpoint 7A company expects its income during the next 10 years to be given by c(t) = 20,000t, for 0 ≤ t ≤ 10. Assuming an annual inflation rate of 5%, what is the present value of this income?

To determine

To calculate: The present value of the income if the expected income of company during next 10 years is given by c(t)=20,000t,for 0t10 and the annual rate of inflation is 5%.

Explanation

Given Information:

The expected income of company during next 10 years is given by c(t)=20,000t,for 0t10 and the annual rate of inflation is 5%.

Formula used:

1. If c(t) represents a continuous income function in dollars per year and the annual rate of inflation is r (in decimal form), then the present income over t years is given by

Present value=0tc(t)ertdt

2. The integration by parts udv=uvvdu.

Where, u and v are function of x.

eaxdx=eaxa+C

Calculation:

Consider the provided information that the expected income of company during next 10 years is given by c(t)=20,000t,for 0t10 and annual rate of inflation is 5%, i.e. r=0.05

Now calculate the present value by using formula, Present value=0tc(t)ertdt,

01020000te0.05tdt=20000010te0.05tdt

Consider the indefinite integral 010te0.05tdt

Here,

dv=e0.05tdt and u=t

First find v,

dv=e0.05tdtdv=e0.05tdt

On further solving,

v=e0.05t0.05 …… (1)

Find du:

u=t

Differentiate both side with respect t;

dudt=dtdtdudt=1

And,

du=1dt …… (2)

Apply integration by parts formula and substitute equation (1) and (2) in udv=uvvdu,

te0

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