Question

A \(\$ 1000\) investment earns interest at a rate of \(8 \%\) per year, compounded quarterly. a) Write an equation for the value of the investment as a function of time, in years. b) Determine the value of the investment after 4 years. c) How long will it take for the investment to double in value?

Short answer

a) \[ A(t) = 1000 \left(1.02\right)^{4t} \] b) \[ A(4) \approx 1372.80 \] c) \[ t \approx 8.75 \] years

Step-by-step solution

Understanding the Compound Interest Formula

The compound interest formula is given by \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where:- \(A\) 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 (decimal).- \(n\) is the number of times that interest is compounded per year.- \(t\) is the time the money is invested for, in years.

Identifying the Given Values

Given:- Principal amount \(P = 1000\)- Annual interest rate \(r = 8\% = 0.08\)- Quarterly compounding means \(n = 4\)- Time in years \(t\)

Writing the Equation for the Value of the Investment

Using the compound interest formula, the equation for the value of the investment as a function of time \(t\) is:\[ A(t) = 1000 \left(1 + \frac{0.08}{4}\right)^{4t} = 1000 \left(1 + 0.02\right)^{4t} = 1000 \left(1.02\right)^{4t} \]

Determining the Value of the Investment After 4 Years

Substitute \(t = 4\) into the equation:\[ A(4) = 1000 \left(1.02\right)^{4 \times 4} = 1000 \left(1.02\right)^{16} \]Now, calculate \(\left(1.02\right)^{16}\):\[ \left(1.02\right)^{16} \approx 1.3728 \]So,\[ A(4) = 1000 \times 1.3728 \approx 1372.80 \]

Determining the Time for the Investment to Double

For the investment to double, \(A(t) = 2000\). Using the equation:\[ 2000 = 1000 \left(1.02\right)^{4t} \]Divide both sides by 1000:\[ 2 = \left(1.02\right)^{4t} \]Take the natural logarithm on both sides:\[ \ln(2) = \ln\left(\left(1.02\right)^{4t}\right) \]Applying the power rule of logarithms:\[ \ln(2) = 4t \cdot \ln(1.02) \]Now solve for \(t\):\[ t = \frac{\ln(2)}{4 \cdot \ln(1.02)} \]Calculate the values: \[ \ln(2) \approx 0.6931 \]\[ \ln(1.02) \approx 0.0198 \]\[ t \approx \frac{0.6931}{4 \times 0.0198} = \frac{0.6931}{0.0792} \approx 8.75 \]So, it will take approximately 8.75 years for the investment to double in value.

Key concepts

interest rate

Interest rate is the percentage at which your investment grows per period of time. It's crucial to understand that this rate greatly affects the amount of money you will accumulate over time. In this exercise, the annual interest rate is given as 8%. This means if you invest \(1000, at the end of the year, you would earn 8% of \)1000 which is $80. However, interest is compounded quarterly in this case. This means your interest is calculated and added to your investment four times a year. To find the quarterly rate, we divide the annual rate by 4 (since interest is compounded quarterly). So, the quarterly interest rate is \(\frac{8%}{4} = 2% = 0.02\). Each quarter, you earn 2% on your investment.

investment growth

Investment growth refers to how much your initial investment increases over time. In compound interest scenarios, growth doesn't just depend on the initial amount, but also on the interest gained. Each period (quarter, year, etc.), interest is added not just to the initial sum but to the accumulated amount up to that point. This compound effect allows your money to grow exponentially rather than linearly. In our exercise, with a quarterly compound interest rate, the value of the investment after t years is described by the formula: \[ A(t) = 1000 \times \bigg(1 + 0.02\bigg)^{4t} \] This illustrates that the investment's growth accelerates over time due to compounding.

exponential functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions are key when dealing with compound interest because they model the accelerated growth nature of compounding. In our example, the formula \(A(t) = 1000 \times (1.02)^{4t}\) demonstrates an exponential function. Here, \(1.02\) is the base (slightly larger than 1 due to the 2% interest each quarter), and \(4t\) is the exponent, representing the number of compounding periods. Exponential functions have the distinctive property of growing faster as time passes, which is why the value of an investment under compound interest increases significantly over longer periods.

logarithms

Logarithms are the inverse operations of exponentiation. They are used to solve problems where the exponent (time in these problems) is the unknown. To find the time required for an investment to double, we use logarithms. Starting with the formula for compounded growth: \(2000 = 1000 \times (1.02)^{4t}\), we first simplify it to \(2 = (1.02)^{4t}\). Applying the natural logarithm to both sides, we get \(\text{ln}(2) = \text{ln}\big((1.02)^{4t}\big)\). Using the rule \(\text{ln}(a^b) = b \times \text{ln}(a)\), it becomes \(\text{ln}(2) = 4t \times \text{ln}(1.02)\). Finally, solving for \(t\), we find \(t = \frac{\text{ln}(2)}{4 \times \text{ln}(1.02)} \), revealing the time needed for the investment to double.