Exponential Equations: Continuous Compound Interest Application

One of the most common applications of the exponential functions is the calculation of compound and continuously compounded interest. This discussion will focus on the continuously compounded interest application.

The formula for continuously compounded interest, which is different from the compounded interest formula, is:

COMPOUND INTEREST FORMULA

A = Pe^rt

Where A is the account balance, P the principal or starting value, e the natural base or 2.718, r the annual interest rate as a decimal and t the time in years.

Let's solve a few continuously compounded interest problems.

A savings fund is opened with $2750. The fund is compounded continuously with an interest rate of

7 \frac{1}{4}

%. What will be the account balance after 15 years?

Step 1: Identify the known variables.

Remember that the rate must be in decimal form.

A = ? Account balance

P = $2750 Starting value

r = 0.0725 Decimal form

t = 15 No. of years

Step 2: Substitute the known values.

A = Pe^rt

A = 2750e^(0.0725)(15)

Step 3: Solve for A.

A = 2750e^1.0875 Original

A = $8129.36 Simplify

Example 1: Kevin's father wants to open an account with $5000 that will grow to $12,750 in ten years. What is the minimum interest rate the account can have if compounded continuously?

Step 1: Identify the known variables.

Remember that the rate must be in decimal form.

A = $12,750 Account balance

P = $5000 Principal

r = ? Decimal form

t = 10 No. of years

Step 2: Substitute the known values.

A = Pe^rt

12,750 = 5000e^(r)(10)

Step 3: Solve for r.

12,750 = 5000e^10r Original

2.55 = e^10r Divide by 5000

ln 2.55 = ln e^10r Take ln

ln 2.55 = 10r Inverse

$\frac{\ln 2.55}{10} = r$ Divide by 10

$0.094 \approx r \approx 9.4 %$

Example 2: Katrina started an account with all the money she received as graduation gifts. The account had an interest of

8 \frac{1}{3}

% compounded continuously. After exactly four years the account held $13,700. How much did Katrina receive for graduation?

Step 1: Identify the known variables.

Remember that the rate must be in decimal.

A = $13,700 Account balance

P = ? Principal

r = 0.083 Decimal form

t = 4 No. of years

Step 2: Substitute the known values.

A = Pe^rt

13,700 = Pe^(0.083)(4)

Step 3: Solve for P.

13,700 = Pe^0.33 Original

$\frac{13, 700}{e^{0.33}} = P$ Divide by e^0.33

$9816.48 = P

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