Problem 75 - Calculus: Early Transcendentals

Question

Suppose \(\$ 5000\) is invested in a savings account for 10 years ( 120 months), with an annual interest rate of \(r\) compounded monthly. The amount of money in the account after 10 years is given by \(A(r)=5000(1+r / 12)^{120}\) a. Show there is a value of \(r\) in \((0,0.08)-\) an interest rate between \(0 \%\) and \(8 \%-\) that allows you to reach your savings goal of \(\$ 7000\) in 10 years. b. Use a graph to illustrate your explanation in part (a). Then approximate the interest rate required to reach your goal.

Step-by-Step Solution

Answer

Answer: An interest rate between 4% and 5% is approximately needed to reach the savings goal of $7000 in 10 years.

Step 1: We are given the formula for compound interest, \(A(r) = 5000(1 + r/12)^{120}\). Here, A(r) represents the amount of money in the account after 10 years (120 months) at an annual interest rate of r compounded 120 times. #Step 2: Find the values for r = 0 and r = 0.08#

Let's first calculate the amount A(r) for the \(r\) values given in the exercise. We need to calculate the amounts for r = 0 and r = 0.08. When r = 0: \(A(0) = 5000(1 + 0/12)^{120} = 5000(1)^{120} = 5000\) When r = 0.08: \(A(0.08) = 5000(1 + 0.08/12)^{120} \approx 10865.72\) #Step 3: Use the Intermediate Value Theorem to verify there's an r value between 0 and 0.08#

Step 2: Since the amount of money in the account for both r = 0 and r = 0.08 (\(5000 and \)10865.72) brackets our target value of \(7000, and the compound interest function is continuous, we can apply the Intermediate Value Theorem (IVT) to show that there must exist a value of r between 0 and 0.08 such that A(r) = \)7000. #Step 4: Graph the function to approximate the interest rate needed to reach the savings goal#

Now we must graph the compound interest function \(A(r) = 5000(1+r/12)^{120}\) in the domain of r = [0, 0.08] and find the point where the graph intersects with the horizontal line \(A(r) = 7000\). This intersection point will give us the approximate interest rate required to reach our savings goal of $7000 within 10 years. After graphing the function, we can observe that the intersection is roughly between 0.04 and 0.05, which corresponds to an interest rate between 4% and 5%.

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus, especially useful in understanding continuous functions. Imagine a continuous curve—no jumps or breaks—between two points. The theorem posits that if a value lies between these two points, then there must be at least one corresponding input at which the function attains this value.
For our savings account problem, we use the IVT to find an interest rate that makes our savings grow to $7000 in 10 years. Suppose we calculate the account balance at interest rates of 0% and 8%. We find balances of $5000 and $10865.72, respectively. The target balance of $7000 falls between these amounts.
The account balance as a function of the interest rate is continuous; hence by the IVT, there is some interest rate between 0% and 8% for which the balance exactly equals $7000. This demonstrates the power of the theorem in predicting solutions just using boundary conditions.

Savings Account

A savings account is a secure place to store money while growing it over time. Banks pay interest on the balance, serving as an incentive for keeping money in the account. The focus here is on compound interest, where interest is earned on both the initial deposit and the accumulated interest from previous periods.
When money is compounded monthly, as in our example, interest is added to the principal balance each month, thus increasing the base upon which future interest is calculated. Over time, this compound effect can significantly increase the total amount in the account, making it a powerful form of investment.
By understanding the compounding effect, students can not only plan for achieving specific savings goals, like $7000 in 10 years, but also appreciate the benefits of early and regular saving, especially when earning interest compounded at frequent intervals.

Interest Rate Calculation

Calculating interest rate is central to determining how much a savings will grow over time. In our example, the balance after 10 years depends on the initial deposit, the annual interest rate, and the compounding frequency.
For a monthly compounding frequency, the annual rate is divided by 12. The formula used is: \[ A(r) = 5000 \times \left( 1 + \frac{r}{12} \right)^{120} \]To find the necessary interest rate to reach \(7000, we set this equation equal to \)7000 and solve for \( r \). Analytical or numerical methods, along with graphing techniques, help in finding this rate, especially when the equation can't be easily solved by hand.
Graphing the equation gives a visual representation, and the intersection with the target balance line helps estimate the rate needed. The observed range between 4% and 5% indicates potential interest rates that could help achieve our savings goal within the desired timeframe. Hence, the process highlights the compatibility of mathematical methods with financial planning objectives.