Chapter 6.4, Problem 41E

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 where C 0 is the original investment, t is the time in years, r is the annual interest rate (in C = C o ∫ o n c ( t ) e − r t d t 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 o = $300 , 000 , c ( t ) = 15 , 000 t , r = 6 % (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 Here, 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 original investment C0=$300,000, the capitalized annual cost of maintenance c(t)=15,000t and the rate of interest compounded annually r=6%.

Formula used:

The integral of an exponential is:

eaxdx=eaxa+C

Here, C is integration constant.

Calculation:

Recall the expression of capitalized cost of asset.

C=C0+0nc(t)ertdt

Convert r to fraction.

r=6%=6100=0.06

Substitute C0=\$300,000, c(t)=15,000, n=5 and r=0.06.

C=300,000+0515,000te0.06tdt=300,000+15,00005te0.06tdt

Now, consider the integral and solve separately.

te0.06tdt

Assume dv=e0.06t

Integrate,

dv=e0.06tdtv=e0.06t0.06

And assume u=t

(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

Here, 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

Here, 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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