Problem 51 - Calculus and its applications

Question

The capitalized cost, \(c,\) of an asset over its 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 with the formula $$ c=c_{0}+\int_{0}^{L} m(t) e^{-k t} d t $$ where \(c_{0}\) is the initial cost of the asset, \(L\) is the lifetime (in years), \(k\) is the interest rate (compounded continuously), and \(m(t)\) is the annual cost of maintenance. Find the capitalized cost under each set of assumptions. $$ \begin{array}{l} c_{0}=\$ 300,000, k=5 \%, m(t)=\$ 30,000+\$ 500 t, \\ L=20 \end{array} $$

Step-by-Step Solution

Answer

The capitalized cost is approximately $993,095.

Step 1: Understand the Given Formula

We are given the formula for capitalized cost as \( c = c_{0} + \int_{0}^{L} m(t) e^{-k t} d t \). This includes the initial cost \( c_{0} \) and the present value of future maintenance costs over the asset's lifetime \( L \), compounded at interest rate \( k \). The function \( m(t) \) represents the annual maintenance cost.

Step 2: Identify Given Values

From the problem statement, we have \( c_{0} = \\(300,000 \), \( k = 0.05 \), \( m(t) = \\)30,000 + \$500t \), and \( L = 20 \) years. We'll use these values in our calculation.

Step 3: Integrate Maintenance Cost Function for Present Value

The integral \( \int_{0}^{L} m(t) e^{-k t} d t \) becomes \( \int_{0}^{20} (30,000 + 500t) e^{-0.05t} dt \). We need to calculate this integral which includes the term for maintenance cost.

Step 4: Integrate Each Term Separately

The integral can be broken down into two parts: \[ \int_{0}^{20} 30,000 e^{-0.05t} dt + \int_{0}^{20} 500t e^{-0.05t} dt \]. We'll calculate each integral separately and then sum the results.

Step 5: Calculate First Integral

Use substitution to calculate the first integral: \( \int_{0}^{20} 30,000 e^{-0.05t} dt \). The antiderivative of \( e^{-0.05t} \) is \( -20e^{-0.05t} \). Evaluating from 0 to 20 gives \(-600,000 \left[ e^{-1} - 1 \right]\).

Step 6: Calculate Second Integral

For the second integral \( \int_{0}^{20} 500t e^{-0.05t} dt \), use integration by parts. Let \( u = t \) and \( dv = 500 e^{-0.05t} dt \). Compute the antiderivative and the result, evaluate from 0 to 20.

Step 7: Sum the Integrals' Values

Combine both evaluated integrals and add the initial cost \( c_{0} \) to find the total capitalized cost \( c \).

Step 8: Calculate Final Capitalized Cost

Add the value of the initial cost \( c_{0} \) to the sum of the evaluated integrals to get \( c = 300,000 + \text{(sum of integrals)} \). Compute for the final answer.

Key Concepts

Present Value

The concept of present value is fundamental in financial mathematics. It refers to the current worth of a sum of money or stream of cash flows that is expected to be received or paid in the future, discounted at a specified interest rate. This concept is crucial in calculating the capitalized cost of an asset.

In the given problem, the capitalized cost formula includes the present value of the maintenance expenses that occur over time. To determine this, we adjust future maintenance costs to present value using the exponential discount factor, where the interest rate is compounded continuously.

• The exponential term in the formula, $e^{-kt}$, addresses how future costs diminish in present value, due to the interest rate $k$.
• The approach ensures we accurately reflect how receiving $30,000 in the future is valued today.

Understanding present value assists in determining how much future expenses are worth now, making it easier to allocate financial resources efficiently.

Maintenance Cost

Maintenance cost refers to the expenses required for upkeep and repair of an asset throughout its lifetime. In our formulation of capitalized cost, maintenance expenses are modeled by a function, $m(t)$, which encompasses both fixed and variable components.

Here, the annual cost of maintenance is given as $m(t) = 30,000 + 500t$, combining a fixed cost of $30,000$ with an additional amount dependent on time.

• The term $30,000$ represents regular, predictable expenses.
• The variable term $500t$ represents scaling costs, perhaps due to more intensive repairs needed as the asset ages.

These equations allow planners to estimate future maintenance costs effectively, ensuring financial provisions are made to cover these expenses without impacting the firm's cash flow.

Integration by Parts

Integration by parts is a mathematical technique used to integrate products of functions, such as \(u \cdot v\). It transforms the product into a simpler form, often when standard integration techniques are insufficient. This method is derived from the product rule of differentiation.

In the exercise provided, integration by parts is essential for finding the present value of a time-dependent maintenance cost. For the component \(500t \cdot e^{-0.05t}\), we set:

• \(u = t\) which makes \(du = dt\).
• \(dv = 500e^{-0.05t} dt\), which integrates to \(v = -10000e^{-0.05t}\).

Using integration by parts, we simplify the integral into a form where direct evaluation is feasible: \[\int_{0}^{20} 500t \cdot e^{-0.05t} \, dt = \left[-10000te^{-0.05t} \right]_0^{20} - \int_{0}^{20} -10000e^{-0.05t}dt\] This process makes seemingly complex integrations manageable, allowing us to calculate precise capitalized costs for future and variable maintenance expenses.