Problem 20 - Applied Calculus

Question

Your company is considering buying new production machinery. You want to know how long it will take for the machinery to pay for itself; that is, you want to find the length of time over which the present value of the profit generated by the new machinery equals the cost of the machinery. The new machinery costs \(\$ 130,000\) and earns profit at the continuous rate of \(\$ 80,000\) per year. Use an interest rate of \(8.5 \%\) per year compounded continuously.

Step-by-Step Solution

Answer

The machinery will pay for itself in approximately 6.12 years.

Step 1: Understand the Present Value Formula

The present value (PV) of a continuous cash flow can be calculated using the formula: \[ PV = \int_0^T C e^{-rt} \, dt \] where \( C \) is the continuous cash flow (profit rate), \( r \) is the interest rate, and \( T \) is the time period over which we want the present value to equal the cost of the machinery.

Step 2: Set up the Equation

Since we want the present value of profits from the machinery to equal the cost of the machinery, set the present value equal to the machinery cost: \[ \int_0^T 80000 e^{-0.085t} \, dt = 130000 \]

Step 3: Solve the Integral

Calculate the definite integral: \[ \int_0^T 80000 e^{-0.085t} \, dt = \left[ -\frac{80000}{0.085} e^{-0.085t} \right]_0^T \] This simplifies to \[ -\frac{80000}{0.085} e^{-0.085T} + \frac{80000}{0.085} \]

Step 4: Simplify the Equation

Set the simplified equation equal to the machinery cost: \[ -\frac{80000}{0.085} e^{-0.085T} + \frac{80000}{0.085} = 130000 \] Simplify and solve for \( e^{-0.085T} \): \[ e^{-0.085T} = 1 - \frac{130000 \times 0.085}{80000} \]

Step 5: Solve for Time \( T \)

Calculate the right-hand side: \[ e^{-0.085T} = 0.6125 \] Take the natural logarithm of both sides to solve for \( T \): \[ -0.085T = \ln(0.6125) \] \[ T = \frac{-\ln(0.6125)}{0.085} \] Calculate \( T \), which is approximately \( 6.12 \) years.

Key Concepts

Continuous Cash Flow

Continuous cash flow refers to a stream of monetary payments or receipts that occur at every moment over a specified time frame. In business scenarios, particularly in investments, continuous cash flow represents ongoing revenue that a project or asset generates. This concept is essential because it allows financial analysts to determine the present value of these cash flows, using mathematical integrals, to make informed investment decisions.

Consider the problem of deciding how long it will take for the new production machinery to pay for itself. This requires understanding how the machinery earns money continuously at a rate of $80,000 per year. Since this earning is continuous, we use calculus to analyze the flow. By integrating the continuous cash flow over time, we capture the total value generated during that period, adjusted for the interest rate.

In our example, the present value formula for a continuous cash flow is used:

• Continuous flow rate ( C): 80,000
• Integration limits: From 0 to T, where T is the unknown time
• Function inside the integral: an exponential decay due to interest

Calculating this helps in understanding the value of money earned by the machinery before the interest rate impacts its worth.

Interest Rate Calculation

Interest rate calculation, especially for continuous compounding, plays a vital role in evaluating investments and financial returns. In the original problem, an 8.5% interest rate was mentioned, which is compounded continuously. Continuous compounding differs from the typical compounding periods (like annually, semi-annually) and is expressed mathematically using the exponential function.

Here's a deeper look:

• Formula: When money is continuously compounded, the formula used is e raised to the power of the interest rate and time.
• Explanation: Interest compounds constantly, making the money grow faster
• Current example: the interest affects the value of future cash flows, reducing them to their present value states

In the machinery example, the interest rate calculation involves adjusting the future profit expectations to reflect present values. This was done by solving the exponential equation dealing with e, the base of natural logarithms. Calculating the present value using this approach tells us the effective earning from today's perspective.

Definite Integral

A definite integral is a fundamental concept in calculus used to calculate the total area under a curve, which in this context, represents the present value of continuous cash flows over a specific time frame. Within our machinery scenario, solving a definite integral was necessary to determine at what point in time the machinery's profit equals its cost, $130,000.

Let's break this down further:

• Understanding the integral: The integral from 0 to T signifies calculating the cumulated present value from start until time T
• Impacted by interest: Profit amounts at future times are adjusted by the exponential decay that is factored into the integrand e^{-rt}
• Integration result: provides an expression of present value depending on time

The definite integral allowed us to create an equation where we set the output, the present value of accrued profits, equal to the machinery's cost. Solving this integral equation and further algebraic manipulation helped identify when the cost equals the present value, providing essential insights into the financial break-even point for the machinery.