Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Compound Interest A deposit of \(\$ 3000\) reaches a balance of \(\$ 4296.16\) after 6 years. The interest on the account is compounded monthly. What is the annual interest rate for this investment?

Step-by-Step Solution

Answer

The approximate annual interest rate for this investment, compounded monthly, is 4.89%.

Step 1: Identify and Plug Known Values into the Formula

Here, the principal amount (P) = $3000, the future value (A) = $4296.16 and t (time invested in years) = 6. We aim to find 'r' (annual interest rate), keeping in mind that the rate is compounded monthly, therefore, n = 12 in a given year. Substitute these into the formula, \(4296.16 = 3000(1 + r/12)^{12*6}\).

Step 2: Simplify the Equation

By dividing the equation through by 3000, we get: \(1.432053 = (1 + r/12)^{72}\). Since we need to isolate 'r', it's best to get rid of the power of 72 first. We do this by taking the 72nd root of both sides. Doing this, we get \((1.432053)^{1/72} = 1 + r/12\)

Step 3: Solve for 'r'

To isolate 'r', subtract 1 from both sides first. Then, 'r/12' equals to \((1.432053)^{1/72} - 1\). Afterwards, multiply both sides by 12 in order to get 'r' on its own. This gives us an annual rate 'r' of about 0.0489.

Key Concepts

Compound Interest Formula

Understanding the compound interest formula is crucial when dealing with investments and savings. It's the mathematical expression used to calculate the amount of interest earned on an initial sum of money (principal) when the interest is reinvested to earn additional interest. The compound interest is usually higher than simple interest because it takes into account the interest on interest effect.

The general formula for compound interest is: \[ A = P(1 + \frac{r}{n})^{nt} \]
where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (the initial sum of money).
• \(r\) is the annual interest rate (in decimal form).
• \(n\) is the number of times that interest is compounded per year.
• \(t\) is the time the money is invested for, in years.

To solve for any of the variables in the formula, algebraic manipulation may be required, as shown in the exercise where the equation is rearranged to find the annual interest rate, \(r\).

Exponential Equations

Exponential equations are those that include variables in the exponent position. They often appear in real-world scenarios, particularly in financial mathematics where compound interest calculation is essential. The base form of an exponential equation is \[ y = a^{x} \]
with \(y\) being the result, \(a\) the base, and \(x\) the exponent. Solving exponential equations like the one in our exercise often requires taking the nth root or using logarithms to bring down the exponent.

However, the critical step is to isolate the base and its exponent on one side of the equation before any other manipulation. This allows for these mathematical operations to be used effectively. The step-by-step solution of the compound interest problem required the isolation of the term with \(r\) in the exponent, allowing us to proceed to find its value.

Such equations can initially seem intimidating, but by breaking them down into their basic components, and using systematic approaches like roots or logs, they become manageable and solvable.

Financial Mathematics

Financial mathematics encompasses the creation of mathematical models that help in making decisions regarding finance, such as investing, borrowing, lending, and saving. Compound interest calculations are a staple in this field and provide a way for individuals to understand the potential growth of their investments or the future cost of their loans.

In our textbook example, we see financial mathematics in action as we determine the annual interest rate for a compounded investment. It incorporates the principles of time value of money, which implies that one dollar today is worth more than one dollar tomorrow, due to the potential earning capacity.

Financial mathematics relies on formulas and models to predict outcomes, manage risks, and optimize financial strategies. Knowing how to manipulate formulas, like the compound interest formula, is key for students and professionals who want to excel in finance. The practical application of these concepts not only helps in academia but in everyday financial planning and decision-making.