Problem 57 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

COMPOUND INTEREST How quickly will \(\$ 2,000\) grow to \(\$ 5,000\) when invested at an annual interest rate of \(8.5\) if interest is compounded: a. Quarterly b. Continuously

Step-by-Step Solution

Answer

When compounded quarterly, it will take approximately 10.65 years. When compounded continuously, it will take approximately 10.74 years.

Step 1: Understand the formula for compound interest (Quarterly)

The formula for compound interest compounded quarterly is \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] 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 interest is compounded per year, and \(t\) is the number of years the money is invested for.

Step 2: Plug in the known values for (Quarterly)

Given: \(P = 2000\), \(A = 5000\), \(r = 0.085\), \(n = 4\). Plug into the formula: \[ 5000 = 2000 \left(1 + \frac{0.085}{4}\right)^{4t} \]

Step 3: Solve for t (Quarterly)

Divide both sides by 2000: \[ 2.5 = \left(1 + \frac{0.085}{4}\right)^{4t} \] Calculate the base: \[ 2.5 = \left(1 + 0.02125\right)^{4t} \] \[ 2.5 = (1.02125)^{4t} \] Take the natural logarithm of both sides: \[ \ln{2.5} = \ln{(1.02125)^{4t}} \] Use the logarithm power rule: \[ \ln{2.5} = 4t \cdot \ln{1.02125} \] Solve for t: \[ t = \frac{\ln{2.5}}{4 \cdot \ln{1.02125}} \] Calculate t: \[ t \approx 10.65\]

Step 4: Understand the formula for compound interest (Continuously)

The formula for continuous compound interest is \[ A = Pe^{rt} \] where \(A\) is the amount of money accumulated, \(P\) is the principal amount, \(r\) is the annual interest rate, and \(t\) is the time in years.

Step 5: Plug in the known values for (Continuously)

Given: \(P = 2000\), \(A = 5000\), \(r = 0.085\). Plug into the formula: \[ 5000 = 2000e^{0.085t} \]

Step 6: Solve for t (Continuously)

Divide both sides by 2000: \[ 2.5 = e^{0.085t} \] Take the natural logarithm of both sides: \[ \ln{2.5} = 0.085t \] Solve for t: \[ t = \frac{\ln{2.5}}{0.085} \] Calculate t: \[ t \approx 10.74 \]

Key Concepts

Quarterly Compounding

Quarterly compounding means that the interest is calculated and added to the principal four times a year.
This type of compounding results in more growth compared to annual compounding because interest is added more frequently.
For quarterly compounding, the formula is where: extrapolate to With quarterly compounding: moving forward an example.
We plug in our given values, and solve for Once you have the base calculation, take the natural logarithm to easily solve for This mathematical concept can seem complicated but understanding each step simplifies the whole calculation process.

Continuously Compounding

Continuous compounding computes interest infinitely many times per year.
It’s the most frequent form of compounding.
The formula regularly used is: Which includes Euler’s number One we’ve understood the variables we can now substitute our values: other example: similar to apply the natural logarithm to solve new values in to yield the final value for

Natural Logarithm

The natural logarithm is a logarithm based on the number e which is approximately takes in enables calculation converting anything into logarithmic useful when tackling complex take ln and a logarithmic transformation you'll note ln simplifies converting to an easier to manage form useful when solving for variables

An instance provided: how useful logarithmic rule a simple one converting complex calculations

Interest Rate Calculations

The rate at which interest is calculated is key when determining the growth of an investment. Certain steps include dismantling: variables ultimately directly affect
solve this and calculate utilize for ease:
variables: demonstrating thorough calculations showing exact numerical accounts application:
solving becomes seamless
lends key insights and accurate projections essential for financial planning