Problem 47 - College Algebra and Calculus: An Applied Approach

Question

MAKE A DECISION: SCHOLARSHIP FUND You want to start a scholarship fund at your alma mater. You plan to give one \(\$ 18,000\) scholarship annually beginning one year from now and you have at most \(\$ 400,000\) to start the fund. You also want the scholarship to be given out indefinitely. Assuming an annual interest rate of \(5 \%\) compounded continuously, do you have enough money for the scholarship fund?

Step-by-Step Solution

Answer

It is only possible to determine if the funds will suffice for the scholarship after the calculations are done. Depending on the calculated annual interest, a decision can be made whether the scholarship fund is sustainable indefinitely or not.

Step 1: Identifying knowns and unknowns

In this problem, we know the following: Initial Principal amount P \(= \$400,000\), Annual scholarship amount \(= \$18,000\), Annual interest rate in decimal form, r \(= 5/100 = 0.05\), and we want the scholarship fund to continue indefinitely.

Step 2: Formulate the Equations

As we want the scholarship to continue indefinitely, we would want the principal amount to make at least \$18,000 in interest annually. That is we want to solve the for \( A \) in the equation \( A = P e^{rt} \). But as the scholarship starts one year from today, we substitute \( t = 1 \) year into our equation.

Step 3: Calculate Annual Interest

Substituting the values of \( P \), \( r \), and \( t \) in the equation we get - \( A = \$400,000 * e^{0.05*1} \). Solving this, the amount made in a year by the endowment is \( A = \$400,000 * e^{0.05} \). Now we compare this \( A \) with the annual scholarship amount of \( \$18,000 \).

Step 4: Decide and conclude

If the calculated \( A \) is greater than or equal to \( \$18,000 \), then the scholarship can be given out indefinitely. If \( A \) is less than \( \$18,000 \), then the scholarship couldn't be given indefinitely.

Key Concepts

Continuous Compounding

Continuous compounding is a unique financial concept that allows an investment's interest to be calculated and added back to the principal consistently over time. While traditional methods use yearly or monthly periods to compute interest, continuous compounding follows the idea that interest is being calculated infinitely at every possible moment. Such compounding results in exponential growth.

When dealing with continuous compounding, we often use the formula:

• \[ A = Pe^{rt} \]
• Where:
  • \( A \) is the amount of money accumulated after a certain time, including interest.
  • \( P \) is the initial principal balance (the original amount of money).
  • \( r \) is the annual interest rate (in decimal form).
  • \( t \) is the time the money is invested for.

In our scholarship problem, continuous compounding ensures that the fund grows steadily, even within just one year. The formula effectively calculates how much money the initial investment can generate if compounded continuously over a short period.

It highlights how investments can quickly grow over time, making it an important concept for those looking to make informed financial decisions especially related to sustaining funds over the long term.

Annual Interest Rate

An annual interest rate is a key factor to consider when determining growth in investments or savings over time. It represents the percentage of the principal, or the original investment, that an investor can expect to earn over the course of a year. Annual interest rates can be applied to any financial product, from loans to bank savings, and are central to financial planning.

In our example with the scholarship fund, the given annual interest rate is 5%. This means that the fund owner expects to earn 5% of the principal amount of \$400,000 varje year. However, it's critical to convert this annual interest rate to a decimal form \(r = 5/100 = 0.05\) for calculations involving compounding formulas.

Understanding annual interest rates enables you to:

• Evaluate whether your investments are growing at a rate that meets your goals.
• Determine the feasibility of creating sustainable long-term commitments, like an indefinite scholarship fund.

A solid grasp of how annual interest rates function is crucial in determining if the growth from your investment is significant enough to cover annual costs or withdrawals.

Indefinite Scholarship

Creating an indefinite scholarship fund is a way to provide lasting educational opportunities, ensuring that students receive scholarships long into the future. This involves making sure that the amount of interest generated annually from the fund's principal is enough to cover the yearly scholarship amount, without exhausting the principal.

For a scholarship to continue indefinitely, the principal must at least sustain or increase in value year after year. In mathematical terms, this requires the continuous growth of the investment due to interest, not only keeping up with inflation but also covering the annual payouts sought. Here, the interest amount should equal or surpass the annual scholarship amount.

With the given exercise, your aim was to see if the \\(400,000, coupled with a 5% continuous annual interest rate, would indefinitely fund a \\)18,000 annual scholarship. The calculation shows whether the interest earned each year is ready to cover the scholarship grant, without dipping into the fund itself.

This concept not only ensures financial sustainability but also fulfills the educational missions envisioned by the funder, hence safeguarding the scholarship's continuance for generations ahead.