Question

Suppose 5000 dollars 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 \(A(r)=5000(1+r / 12)^{120}\). a. Use the Intermediate Value Theorem to 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 dollars in 10 years. b. Use a graph to illustrate your explanation in part (a); then approximate the interest rate required to reach your goal.

Short answer

Question: Show that there is an interest rate between 0% and 8% that allows a $5000 investment to grow to at least $7000 in 10 years with monthly compounding, and approximate the required interest rate using the Intermediate Value Theorem and a graph. Answer: Using the Intermediate Value Theorem, we find that there must exist an interest rate between 0% and 8% that allows the investment to grow to at least $7000. By visually examining a graph of the function, we can approximate the required interest rate to be around 2.8%.

Step-by-step solution

a. Using the Intermediate Value Theorem

The Intermediate Value Theorem (IVT) states that if a continuous function has one value at one point and another value at another point, then, between those two points, it must take any value between those function values at least once. To apply the IVT, we need to calculate the amount (A(r)) for r = 0, and r = 0.08: 1. For r = 0: \(A(0) = 5000(1 + \frac{0}{12})^{120} = 5000(1)^{120} = 5000\) 2. For r = 0.08: \(A(0.08) = 5000(1 + \frac{0.08}{12})^{120} \approx 10889.29\) Since A(r) is a continuous function, and we have \(A(0) = 5000 < 7000 < A(0.08) \approx 10889.29\), there must exist an interest rate r in the interval (0, 0.08) which allows us to reach our savings goal of 7000 dollars according to IVT. #b. Using a graph to illustrate and approximate the interest rate:

b. Using a graph to illustrate and approximate the interest rate

Create a graph of the function \(A(r) = 5000(1+\frac{r}{12})^{120}\) with r in the interval [0, 0.08]. Plot additionally a horizontal line at the savings goal of 7000 dollars on the same graph. The interest rate required to reach the goal corresponds to the point where the graph of \(A(r)\) intersects the horizontal line at 7000 dollars. By visually examining the graph, we can approximate the interest rate required to meet the savings goal. You can also use a calculator or software to find a more accurate intersection point. From the graph, we can approximate the required interest rate (r) to be around 2.8%. Note that this is an approximation; you may use more advanced mathematical tools to find a more accurate interest rate.

Key concepts

Continuous Function

A continuous function is essential in understanding mathematical concepts like the Intermediate Value Theorem (IVT). With a continuous function, you can imagine it as a smooth, unbroken curve on a graph. This means there are no gaps, jumps, or breaks between any two points.

For example, in our case, the function \( A(r) = 5000(1 + \frac{r}{12})^{120} \) describes the amount of money in a savings account after 10 years, depending on the interest rate \( r \). This function is continuous because any small change in \( r \) leads to a small change in \( A(r) \). No sudden large jumps or drops can occur in how the savings amount changes as the interest rate changes.

Continuous functions are important because they allow the use of the IVT. This theorem tells us that if you have two function values – like our \( A(0) = 5000 \) and \( A(0.08) \approx 10889.29 \), any value between these two can be reached by some \( r \) value in the interval (0, 0.08). In our case, this means there's definitely some interest rate between 0% and 8% that grows your savings to exactly $7000 in 10 years.

Compounded Interest

Compound interest is a compelling concept that describes the process of earning interest not only on your initial principal but also on the accumulated interest in previous periods. It essentially means you earn "interest on interest."

In our example, the interest is compounded monthly. This means each month, the bank calculates interest on both the initial deposit and the interest that has already been added to it. The formula for compound interest, \( A(r) = P\left(1 + \frac{r}{n}\right)^{nt} \), helps us calculate how much money will be in our savings account over time.

Here's how it works for our savings problem:

• The initial deposit (principal), \( P \), is \(5000.
• The annual interest rate, \( r \), is what we are trying to find within a certain range, compounded monthly.
• The number of times the interest is compounded per year, \( n \), is 12 because it's compounded monthly.
• The number of years, \( t \), is 10.

Using these, compound interest allows the savings to grow to achieve the desired future value of \)7000 in 10 years at a specific rate.

Savings Account

A savings account is a secure place where people can deposit their money to earn interest over time. It's a simple yet powerful financial tool for growing savings due to the benefits it offers in terms of added interest income.

Typically, banks offer different interest rates on savings accounts, and interest might be compounded over various periods: daily, monthly, or annually. A critical advantage is that it's low-risk, which is essential for anyone new to saving or looking to preserve their wealth over time.

In our scenario, a savings account is used to invest $5000 with interest compounded monthly for 10 years. It's crucial to look at the interest rate offered by the bank, as this determines how much the savings will grow. By using tools such as the Intermediate Value Theorem and graphs, we can estimate the necessary interest rate to achieve a specific savings goal – in this case, $7000.

Knowing how savings accounts and interest rates work allows you to make informed decisions about where to put your money and how to plan for future financial goals.