Problem 39 - Calculus: Early Transcendental Functions

Question

One of the authors bought a set of basketball trading cards in 1985 for \(\$ 34 .\) In \(1995,\) the "book price" for this set was \(\$ 9800 .\) Assuming a constant percentage return on this investment, find an equation for the worth of the set at time \(t\) years (where \(t=0\) corresponds to 1985 ). At this rate of return, what would the set have been worth in \(2005 ?\)

Step-by-Step Solution

Answer

The set of basketball cards bought in 1985 would be worth approximately \$574,717.72 in 2005.

Step 1: Determine the initial value and the future value

The initial value or the price of the set in 1985 is \$34. It's given that the price of the set in 1995 was \$9800, which will be used for our future value. Therefore, \( P_0 = \$34 \) and \( P(10) = \$9800 \), since in this case, \( t = 10 \) years from 1985.

Step 2: Substitute and solve for the rate of return (r)

Substitute \( P_0 \), \( P(t) \), and \( t \) into the exponential growth formula to find the annual rate \( r \). \( \$9800 = \$34 * (1 + r)^{10} \) can then be solved for \( r \). Solve these for \( r \).

Step 3: Solve for r

To solve for \( r \), follow these steps: \ 1. Divide both sides of the equation by 34, resulting in \( 288.235294 = (1 + r)^{10} \). \ 2. Take the 10th root of both sides to isolate \( 1 + r \), resulting in \( 1 + r = 2.4674 \). \ 3. Subtract 1 from both sides to solve for \( r \), resulting in \( r = 1.4674 \) or 146.74%.

Step 4: Determine the value of the set in 2005

Now that the rate of return, \( r \) is known, use this and the initial investment to calculate the future value of the investment in 2005 which is 20 years from 1985. Substitute, \( P_0 = \$34 \), \( r = 1.4674 \), and \( t = 20 \) years into the exponential growth formula to find the value of the set in 2005: \( P(20) = \$34 * (1 + 1.4674)^{20} \).

Step 5: Calculate the value of the set in 2005

Perform the calculation and round the result to the nearest cent: \( P(20) = \$34 * (2.4674)^{20} = \$574717.72 \) approximately.

Key Concepts

Investment Return Rate

The investment return rate is a crucial concept for anyone looking to understand the potential growth of their funds over time. It essentially calculates how much an investment grows annually, taking into account the compound interest effect. In the provided exercise, the return rate is derived from the historical growth of a basketball trading card set.

For students seeking clarity, here's how to approach it: starting with an initial amount (the price paid for the asset), compare it to what it's worth after a certain number of years. The formula to find the annual rate of return when dealing with compound interest is:\[ P(t) = P_0 \times (1 + r)^t \]where \(P(t)\) is the future value after \(t\) years, \(P_0\) is the initial investment, and \(r\) is the rate of return. By rearranging this formula, you can solve for \(r\), giving you the percentage rate at which the investment grows each year.

Exponential Functions

Exponential functions are mathematical expressions that model situations where a quantity grows or decays at a rate proportional to its current value. This characteristic makes them ideal for representing scenarios like population growth, radioactive decay, and, as in our exercise, the appreciation of an investment over time.

An exponential function is generally expressed as:\[ f(x) = a \times b^x \]where \(a\) is the initial amount, \(b\) is the base that represents the growth factor, and \(x\) is the exponent that stands for time passed or number of periods. In the context of investment return, the value of an asset over time can be modeled with an exponential function where the base is \((1 + r)\), with \(r\) being the rate of growth or return.

Future Value Calculation

Understanding the concept of future value is key to making informed investment decisions. The future value calculation involves determining what a sum of money invested today will be worth at a specific point in the future when it has been subjected to compounded interest.

The general formula for calculating the future value is:\[ FV = PV \times (1 + r)^n \]where \(FV\) is the future value, \(PV\) is the present value or initial investment amount, \(r\) is the annual interest rate (expressed as a decimal), and \(n\) is the number of periods over which the investment grows. In our exercise, the future value calculation helps predict the appreciation of the basketball trading card set based on its exponential growth over a 20-year span, considering the annual return rate calculated previously.