Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use logarithms to solve each problem. How long will it take an investment of $$\$ 2000$$ to double if the investment earns interest at the rate of \(9 \% /\) year compounded monthly?

Step-by-Step Solution

Answer

It will take approximately \(7.78\) years for the investment to double at an interest rate of \(9\%\) compounded monthly.

Step 1: Write down the compound interest formula

We will use the following formula: \(A = P(1 + \frac{r}{n})^{nt}\)

Step 2: Substitute the given values into the formula

We want the investment to double, so we let \(A = 2P\), which is \(4000\). We are given \(P=2000\), \(r=0.09\) (since \(9\% = 0.09\)), and \(n=12\) (since interest is compounded monthly). Substituting these values, we have: \[ 4000 = 2000 \left(1 + \frac{0.09}{12}\right)^{12t} \]

Step 3: Solve for t using logarithms

To solve for t, we first need to isolate the exponential part. Divide both sides by 2000: \[ 2 = \left(1 + \frac{0.09}{12}\right)^{12t} \] Now, take the natural logarithm of both sides: \[ \ln(2) = \ln\left(\left(1 + \frac{0.09}{12}\right)^{12t}\right) \] Using the logarithmic property \(\ln(a^b) = b\ln(a)\), we can rewrite the equation as follows: \[ \ln(2) = 12t\ln\left(1 + \frac{0.09}{12}\right) \] Finally, we can find t by dividing both sides by \(12\ln\left(1 + \frac{0.09}{12}\right)\): \[ t = \frac{\ln(2)}{12\ln\left(1 + \frac{0.09}{12}\right)} \]

Step 4: Calculate the value of t

Using a calculator or appropriate software, calculate the value of t: \[ t \approx \frac{\ln(2)}{12\ln\left(1 + \frac{0.09}{12}\right)} \approx 7.78 \] So, it will take approximately \(7.78\) years for the investment to double at an interest rate of \(9\%\) compounded monthly.

Key Concepts

Compound Interest

Compound interest is a way of calculating interest where the interest earned over time is added back to the principal amount, thus forming a new principal.
This means that from this new principal, interest is earned once again, creating a cycle of increasing the total amount.
Using the compound interest formula, we can calculate how much an investment grows over a certain period of time.Here's a quick breakdown of the formula used:

• \(A = P(1 + \frac{r}{n})^{nt}\)
• \(A\) is the amount of money accumulated after \(n\) years, including interest.
• \(P\) is the principal amount (the initial amount of money).
• \(r\) is the annual interest rate (in decimal form).
• \(n\) is the number of times interest is compounded per year.
• \(t\) is the number of years the money is invested for.

Understanding this formula helps investors predict how long it will take to reach their financial goals. In our example, by substituting the known values, we find how long it takes for a $2000 investment to double at a 9% interest rate compounded monthly.

Exponential Equations

Exponential equations involve variables appearing as exponents. In this context of compound interest, the growth of the investment is modeled using an exponential equation.
When solving exponential equations, logarithms are our go-to tools as they allow us to "unwrap" the exponent and solve for the unknown variable more easily.
When our exponential equation,

• \(2 = \left(1 + \frac{0.09}{12}\right)^{12t}\)

becomes tricky due to the exponent \(12t\), logarithms help simplify:

• Take the natural logarithm of both sides to isolate \(t\):
• \(\ln(2) = 12t \cdot \ln\left(1 + \frac{0.09}{12}\right)\)

This transforms the original complex exponent to a more manageable equation for solving \(t\). Using the property \(\ln(a^b) = b\ln(a)\), finding \(t\) becomes a matter of simple division. As seen, exponential equations often represent real-world growth scenarios, like our doubling investment problem.

Mathematical Modeling

Mathematical modeling involves using mathematical structures to represent real-world situations. It helps in predicting future outcomes or analyzing complex systems using mathematical equations.
In finance, mathematical modeling is crucial for understanding investment growth, risk management, and financial planning.
For our exercise, modeling the compound interest scenario allowed us to predict the time needed for an investment to double.
Here, the compound interest formula served as the core model to describe the doubling time of our initial investment.
The act of substituting actual values such as the initial principal, interest rate, and compounding frequency, allows us to see beyond mere numbers.

• By solving these equations, we can make informed predictions.
• Models help in visualizing the relationship between different variables.
• They are essential for strategic financial decisions.

Thus, mathematical modeling is not just about equations, but about crafting a blueprint of the financial landscape to navigate future possibilities.