Problem 11 - Applied Calculus

Question

German TV and radio users pay an annual license fee of 200 euros. In 2008, the German poet Friedrich Schiller, who died in \(1805,\) was sent reminders to pay his license fee. \(^{5}\) (The reminders were sent to an elementary school named after Schiller, despite a teacher pointing out that "the addressee is no longer in a position to listen to the radio or watch television".) Assume the license fee had been charged at a continuous rate of 200 euros per year since 1805 and that the continuous interest rate was \(3 \%\) per year over this period. If Schiller were charged the license fee, with interest, since his death, how much would he have owed by \(2008 ?\)

Step-by-Step Solution

Answer

By 2008, Schiller would have owed approximately 318 million euros.

Step 1: Identify the Time Period

First, calculate the time period from Schiller's death in 1805 to the year 2008 when the reminders were sent. This is a total of 2008 - 1805 = 203 years.

Step 2: Understand the Continuous Compounding Formula

The formula for continuous compounding is given by \( A = Pe^{rt} \) where \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount (initial fee), \( r \) is the annual interest rate (as a decimal), and \( t \) is the time in years.

Step 3: Set up the Variables

In this exercise, the principal \( P \) is the annual fee of 200 euros, the rate \( r \) is the continuous annual interest rate of 3% or \( 0.03 \) in decimal, and the time \( t \) is 203 years. However, since we want the accumulated value over continuous contributions, this turns into a more complex integral problem.

Step 4: Use Integration for Continuous Contributions

To find the total amount owed with continuous contributions and interest over 203 years, integrate the continuous compounding formula. Use \( A(t) = \int_0^{203} 200 e^{0.03(203-x)} \, dx \) where \( x \) is the time the payment was made. Evaluate this integral to find the total amount owed.

Step 5: Compute the Integral

Solving the integral \( \int_0^{203} 200 e^{0.03(203-x)} \, dx \) leads to \( A(t) = 200 \left[ \frac{e^{0.03(203-x)}}{0.03} \right]_0^{203} \). Evaluating this expression, you find \( A = 200 \times \left( \frac{e^{0.03 \times 203} - 1}{0.03} \right) \).

Step 6: Calculate the Final Amount

Compute \( e^{0.03 \times 203} \) and subtract 1, divide by 0.03, and multiply by 200 to find the final amount owed. Using a calculator or software: The computed value is \( A \approx 318,265,523.68 \) euros.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on the accumulation of quantities, such as areas under curves, and is pivotal when dealing with continuous processes over time. In the context of continuous compounding, integral calculus allows us to account for the ongoing accumulation of interest on continuously charged fees. When a fee, like the license mentioned, is charged continuously, calculating its accumulation over an extended period becomes an integration problem.For our exercise, to find how much would've been owed due to continuous charging, we express the total amount as an integral. We set this up as \( A(t) = \int_0^{203} 200 e^{0.03(203-x)} \, dx \). This expression helps us determine the accumulative nature of continuously imposed fees with compound interest, reflecting the rate at which money grows when it constantly accrues interest.Through solving this integral, integral calculus quantifies how input conditions like continuous payment rates and interest compound over time, providing a complete understanding of the financial implications involved.

Exponential Growth

Exponential growth describes how quantities increase at rates proportional to their current size, common in populations and finance. In continuous compounding, the interest accrued grows exponentially over time, significantly increasing the future value of payments or investments.In the Schiller exercise, the formula for exponential growth \( A = Pe^{rt} \) quantifies how continuously compounded interest affects continuously charged fees. Here, exponential growth reflects the ever-increasing total driven by a steady interest rate of 3% per year applied continuously over 203 years.This exponential growth allows us to understand why the accumulated amount by 2008 results in a remarkably high figure, exemplifying how even small, consistent growth rates can lead to substantial outcomes over long periods.

Continuous Interest Rate

A continuous interest rate represents a rate of growth that continuously compounds, unlike standard rates which might compound annually or semi-annually. This concept is crucial when dealing with ongoing investments or fees, providing an accurate measure for growth that reflects real-world scenarios where compounding happens at every instant.In our scenario, we assume a continuous interest rate of 3%. This means that every infinitesimally small amount of money contributing towards the accumulated fee itself earns interest at every point in time. The use of the continuous interest rate is why we apply the exponential function \( e \) (Euler's number) in the formula \( A = Pe^{rt} \) for calculating the total owed by Schiller.The continuous interest rate models financial behavior realistically, showing more accurate projections of growth over extensive periods, crucial for understanding long-term financial obligations or investments.