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Chapter 5: Problem 38

Capitalized cost. The capitalized cost, \(c,\) of an asset for an unlimited lifetime is the total of the initial cost and the present value of all maintenance expenses that will occur in the future. It is computed by the formula $$c=c_{0}+\int_{0}^{\infty} m(t) e^{-r t} d t$$ where \(c_{0}\) is the initial cost of the asset, \(r\) is the interest rate (compounded continuously), and \(m(t)\) is the annual cost of maintenance. Find the capitalized cost under each set of assumptions. $$c_{0}=\$ 700,000, \quad r=5 \%, \quad m(t)=\$ 30,000$$

Short Answer

Expert verified
The capitalized cost is \$1,300,000.

Step by step solution

01

Write down the given information

Identify and write down the values given in the problem. Here, the initial cost of the asset, \(c_0\), is \( \$700,000\), the interest rate, \(r\), is \(5\%\), and the annual maintenance cost, \(m(t)\), is \( \$30,000\).
02

Write the formula for capitalized cost

The formula for capitalized cost is: \[ c = c_{0} + \int_{0}^{\infty} m(t) e^{-r t} \mathrm{d} t \]
03

Set up the integral with the given values

Substitute the given values into the formula: \[ c = 700,000 + \int_{0}^{\infty} 30,000 e^{-0.05 t} \mathrm{d} t \]
04

Evaluate the integral

Use the fact that the integral of \(e^{-r t}\) with respect to \t\ is \( \int e^{-r t} \mathrm{d} t = -\frac{1}{r} e^{-r t} \). Therefore: \[ \int_{0}^{\infty} 30,000 e^{-0.05 t} \mathrm{d} t = 30,000 \left[-\frac{1}{0.05} e^{-0.05 t} \right]_{0}^{\infty} \]This simplifies to: \[ 30,000 \left[-20 e^{-0.05 t} \right]_{0}^{\infty} \]
05

Compute the definite integral

Evaluate the expression by calculating the limits: \[ 30,000 \left[-20 \left(e^{-0.05 \infty} - e^{-0.05 \cdot 0}\right)\right] \] Recall that \( e^{-0.05 \cdot \infty} = 0 \) and \( e^{-0.05 \cdot 0} = 1 \), so this simplifies to: \[ 30,000 \left[-20 \left(0 - 1\right)\right] = 30,000 \left[20\right] = 600,000 \]
06

Combine the initial cost and the total discounted maintenance cost

Add the initial cost of the asset and the present value of all maintenance expenses: \[ c = 700,000 + 600,000 = 1,300,000 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

present value
Present value is a fundamental concept in finance that allows us to find out the value of a sum of money in today's terms that we will receive or pay in the future. The reason we need to calculate the present value is that money has a time value; meaning that a sum of money today is worth more than the same sum in the future due to its potential earning capacity. To calculate the present value, we use the formula:

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