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$$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Time } \\ \text { in months } \end{array} & \text { Savings balance } \\ \hline 0 & \$ 1750.00 \\ \hline 1 & \$ 1771.00 \\ \hline 2 & \$ 1792.25 \\ \hline 3 & \$ 1813.76 \\ \hline 4 & \$ 1835.52 \\ \hline 5 & \$ 1857.55 \\ \hline & \\ \hline \end{array} $$ An investment: You have invested money in a savings account that pays a fixed monthly interest on the account balance. The following table shows the account balance over the first 5 months. a. How much money was originally invested? b. Show that the data are exponential and find an exponential model for the account balance. c. What is the monthly interest rate? d. What is the yearly interest rate? e. Suppose that you made this investment on the occasion of the birth of your daughter. Your plan is to leave the money in the account until she starts college at age 18. How large a college fund will she have? f. How long does it take your money to double in value? How much longer does it take it to double in value again?

Step by step solution

01

Identifying Initial Investment

The initial investment is the account balance at 0 months, which can be directly obtained from the table. Thus, the original investment is $1750.00.

02

Recognizing Exponential Growth

To show the data are exponential, check if there is a constant ratio between the consecutive savings balances. Calculate the ratio of each consecutive month:\( \text{Ratio from Month 0 to 1} = \frac{1771.00}{1750.00} \approx 1.012 \)\( \text{Ratio from Month 1 to 2} = \frac{1792.25}{1771.00} \approx 1.012 \)Continue in this manner, confirming that each ratio is approximately 1.012, indicating exponential growth.

03

Formulating the Exponential Model

The exponential model formula is given by: \( A(t) = A_0 \cdot (1 + r)^t \)where \(A_0\) is the initial amount (\$1750.00), \(r\) is the monthly interest rate, and \(t\) is the time in months. Using the ratio found, the model becomes \( A(t) = 1750.00 \cdot 1.012^t \).

04

Calculating the Monthly Interest Rate

The monthly interest rate can be directly derived from the ratio calculated previously. Thus, the monthly interest rate \( r \) is approximately 1.2%.

05

Calculating the Yearly Interest Rate

To find the yearly interest rate, use: \( (1 + r_{monthly})^{12} - 1 \)Substituting the monthly rate gives:\( (1.012)^{12} - 1 \approx 0.1542 \),which converts to a yearly interest rate of 15.42%.

06

Calculating Future Balance at Age 18

Calculate the time from birth until age 18 in months (18 years \( \times \) 12 months = 216 months) and use the model:\( A(216) = 1750.00 \cdot 1.012^{216} \approx 20514.34 \).Thus, the college fund will be approximately \$20514.34.

07

Determining Time to Double the Investment

To find when the investment doubles, solve for \(t\) when \(A(t) = 2 \times A_0\).Set up the equation:\( 2 \times 1750 = 1750 \cdot 1.012^t \)and solve for \(t\):\( 2 = 1.012^t \).Take the logarithm to solve:\( t = \frac{\ln(2)}{\ln(1.012)} \approx 58.7 \text{ months} \).The investment doubles again after an additional 58.7 months.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interest Rate Calculation

Understanding how to calculate interest rates is crucial when dealing with exponential growth in investments. Interest rates dictate how your money grows over time, and can be expressed on a monthly or yearly basis. In our exercise, we determined the monthly interest rate by observing the ratio between the savings balances of consecutive months. For instance, if in the first month, the balance increased from \(1750 to \)1771, we calculate the month-over-month growth ratio as \( \frac{1771}{1750} \approx 1.012 \). This means an increase of approximately 1.2%. To calculate the annual rate, you need to combine these monthly growths over 12 periods. The formula used is \((1 + r_{monthly})^{12} - 1 \). Applied to our example, \((1.012)^{12} - 1 \approx 0.1542 \), resulting in an annual interest rate of approximately 15.42%. Knowing these calculations enables you to better plan and understand how investments can grow over different timeframes.

Investment Doubling Time

The concept of doubling time is essential in investment growth as it tells you how long it will take for your investment to grow to twice its original amount. By using the exponential growth model, we can determine this time using logarithms. For this calculation, set your future value to double your initial investment, so \( A(t) = 2 \times A_0 \). In our exercise, this means solving \( 3500 = 1750 \times 1.012^t \). Using logarithms, we rearrange to find \( t = \frac{\ln(2)}{\ln(1.012)} \), which approximately equals 58.7 months. This means that under the given interest rate, your investment will double in just under 5 years (or 58.7 months). The longer it takes for an investment to double, the lower the interest rate, showcasing how critical understanding of doubling time is in financial planning.

Exponential Model Formulation

The exponential model is a key aspect when describing growth processes like investments. This model takes into account the initial amount, the interest rate, and the time over which the investment grows. The general form is \( A(t) = A_0 \cdot (1 + r)^t \), where:

• \( A(t) \) is the amount at time \( t \)
• \( A_0 \) is the initial principal amount
• \( r \) is the interest rate per period (monthly in our example)
• \( t \) is the number of periods that have elapsed

In our exercise, we derived \( A(t) = 1750 \cdot 1.012^t \). This shows how applying the interest rate consistently over time results in exponential growth of the investment. An exponential model helps you various scenarios, adjust for different rates or investment lengths, and easily visualize how changes can affect growth targets.

Future Value Calculation

Future value calculation helps project how much your investment will be worth at a specific point in time. Using the exponential model, you can predict the potential growth over a set number of periods. To calculate future value, plug in your desired period (in months or years) into the exponential model. From our exercise, we projected the value of an account over 18 years—which is 216 months. Using the model \( A(216) = 1750 \cdot 1.012^{216} \), we determined the future value to be approximately \$20514.34. This calculation shows the power of compounding interest over a long period. By planning your investment and regularly using future value calculations, you can set realistic goals for savings, debt payoffs, and long-term financial planning.