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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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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 = $ 650 , 000 , c ( t ) = 25 , 000 ( 1 + 0.08 t ) , r = 12 %

(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=$650,000, the capitalized annual cost of maintenance c(t)=25,000(1+0.08t) and the rate of interest compounded annually r=12%.

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=12%=12100=0.12

Substitute C0=$650,000, c(t)=25,000(1+0.08t), n=5 and r=0.12.

C=650,000+05(25,000)(1+0.08t)te0.12tdt=650,000+25,00005(1+0.08t)te0.12tdt

Now, consider the integral and solve separately.

(1+0.08t)te0.12tdt

Assume dv=e0.12t

Integrate,

dv=e0.12tdtv=e0.12t0

(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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