Chapter 6.4, Problem 39E

Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Capitalized Cost In Exercises 39–42, the capitalized cost C of an asset is given by C = C 0 + ∫ 0 n c ( t ) e − r t   d t where C 0 is the original investment, t is the time in years, r is the annual interest rate (in decimal form) compounded continuously, and c(t) is the annual cost of maintenance (in dollars). Find the capitalized cost of an asset (a) for 5 years, (b) for 10 years, and (c) forever. C 0 = $650 , 000 , c ( t ) = 25 , 000 , r = 10 % (a) To determine To calculate: The capitalized cost of asset for 5 years. If the capitalized cost is given by C=C0+0nc(t)ertdt. Where C0 is the original investment, c(t) is the capitalized annual cost of maintenance, r is the rate of interest compounded annually and t is the time. Explanation Given Information: The provided values are given below as, C0=$650,000c(t)=25,000r=10%

Formula used:

The integral of an exponential is defined as,

eaxdx=eaxa+C

Here, C is integration constant.

Calculation:

Consider the provided formula, C=C0+0nc(t)ertdt

Convert r to fraction.

r=10%=10100=0.1

Substitute the values C0=\$650,000, c(t)=25,000, n=5 and r=0.1 in given formula and apply the exponential formula,

C=650,000+0525,000e0

(b)

To determine

To calculate: The capitalized cost of asset for 10 years. If the capitalized cost is given by:

C=C0+0nc(t)ertdt. Where C0 is the original investment, c(t) is the capitalized annual cost of maintenance, r is the rate of interest compounded annually and t is the time.

(c)

To determine

To calculate: The capitalized cost of asset for forever. If the capitalized cost is given by:

C=C0+0nc(t)ertdt. Where C0 is the original investment, c(t) is the capitalized annual cost of maintenance, r is the rate of interest compounded annually and t is the time.

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