Problem 30 - Algebra 2

Question

Savings A student wants to save \(\$ 8000\) for college in five years. How much should be put into an account that earns 5.2\(\%\) annual interest compounded continuously?

Step-by-Step Solution

Answer

The student should put approximately \$6613.15 into the account.

Step 1: Identify the variables

From the problem, A (future amount) is \(\$8000\), r (interest rate) is \(5.2\% = 0.052\), and t (time) is 5 years. We are solving for P (principal or the amount to be invested). Thus, P = ?, A = \$8000, r = 0.052, t = 5.

Step 2: Insert the values into the formula

Substitute the given values into the formula: \(P = 8000 / e^{0.052 \times 5}\).

Step 3: Calculate

Deliver the calculation. Remember Euler's number e is approximately 2.71828. So, \(P = 8000 / 2.71828^{0.052 \times 5}\). After doing the maths, the approximate value for P is found.

Key Concepts

Future Value Calculation

Understanding how to calculate the future value of an investment is essential when planning for financial goals. The future value calculation answers the question of how much a certain amount of money today will be worth in the future when it has been subject to a specific interest rate over a period of time. In our example, we aim to find the amount needed to invest today to achieve a future value of $8000 in five years.

When interest is compounded continuously, the future value calculation uses the formula:

• Future Value (A) = Principal Amount (P) \times e^{rt},

where \(e\) is the base of the natural logarithm, equivalent to approximately 2.71828, \(r\) is the interest rate, and \(t\) is the time period in years. By rearranging this formula, we can solve for the principal amount \(P\) when given the desired future value \(A\). Calculating this helps us determine how much we need to save today to reach our future goal.

Interest Rate

The interest rate is a powerful tool that determines how much your investment will grow over time. In the context of continuous compounding, the interest rate is expressed as a decimal. Thus, an interest rate of 5.2% would be converted to 0.052. This conversion is essential for accurate calculations.

• Interest rates can be annual rates, and when compounded continuously, affect the growth of the principal exponentially over time.

A higher interest rate will increase the growth of your investment more significantly than a lower rate. In the example, using 5.2% allows us to calculate how quickly and efficiently the principal will grow to meet the future savings objective over five years. Understanding how the interest rate affects your future calculations is crucial for making informed financial decisions.

Principal Amount

The principal amount is the initial sum of money that you invest or save. In our scenario, it's the amount you need to deposit today to achieve a desired future balance of $8000 in five years, given continuous compounding at a 5.2% interest rate.

• This initial investment is the key factor you need to determine, and it’s crucial for effective financial planning.
• The principal amount is calculated by rearranging the future value formula. We solve for \(P\) by using \(P = \frac{A}{e^{rt}}\).

Calculating the correct principal prevents over or under-saving for future financial goals, ensuring that your investments are both efficient and effective in meeting the desired future value.

Exponential Function

The exponential function is a mathematical expression that represents exponential growth, which is a powerful concept in finance, especially in continuous compounding. In our example, it is denoted by \(e^{rt}\), where \(e\) is Euler's number, approximately 2.71828.

• This expression models how the principal amount grows exponentially over time.
• The exponential function is used to describe processes that increase at a rate proportional to their values, such as compound interest.

Continuous compounding utilizes this exponential growth model, making it a potent method for growing savings. As \(t\) (time) and \(r\) (interest rate) increase, the value of \(e^{rt}\) grows, showing the powerful effect of time on the future value of an investment.