Use logarithms to solve the problem.
How long will it take $15,000 to grow to $18,000 if the investment earns interest at the rate of 2%/year compounded monthly? (Round your answer to two decimal places.)


To solve this, we'll use the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:
- $ A $ is the amount of money after $ t $ years (future value),
- $ P $ is the principal amount (initial investment),
- $ r $ is the annual interest rate (in decimal form),
- $ n $ is the number of times interest is compounded per year,
- $ t $ is the time in years.

### Given values:
- $ A = 18,000 $ (future value),
- $ P = 15,000 $ (initial investment),
- $ r = 0.02 $ (2% interest rate per year),
- $ n = 12 $ (compounded monthly).

### Step 1: Plug the values into the compound interest formula.

\[
18,000 = 15,000 \left(1 + \frac{0.02}{12}\right)^{12t}
\]

### Step 2: Simplify the equation.

Calculate the monthly interest rate:

\[
1 + \frac{0.02}{12} = 1 + 0.0016667 = 1.0016667
\]

Now the equation becomes:

\[
18,000 = 15,000 \cdot (1.0016667)^{12t}
\]

### Step 3: Isolate the exponential term.

Divide both sides by 15,000:

\[
\frac{18,000}{15,000} = (1.0016667)^{12t}
\]

\[
1.2 = (1.0016667)^{12t}
\]

### Step 4: Take the logarithm of both sides.

Using the natural logarithm (ln):

\[
\ln(1.2) = \ln((1.0016667)^{12t})
\]

Using the property $ \ln(a^b) = b \cdot \ln(a) $:

\[
\ln(1.2) = 12t \cdot \ln(1.0016667)
\]

### Step 5: Solve for $ t $.

\[
t = \frac{\ln(1.2)}{12 \cdot \ln(1.0016667)}
\]

Calculating this:

1. $\ln(1.2) \approx 0.1823$
2. $ \ln(1.0016667) \approx 0.0016661 $

\[
t = \frac{0.1823}{12 \cdot 0.0016661} \approx \frac{0.1823}{0.0199932} \approx 9.12
\]

### Answer:
It will take approximately **9.12 years** for the investment to grow to $18,000.

Would you like more details on any step?

---

#### Related Questions:
1. How does the frequency of compounding affect the growth of an investment?
2. What would the time be if the interest rate were 3% instead of 2%?
3. How would the calculation change if compounded quarterly instead of monthly?
4. How can the formula be applied to calculate continuous compounding?
5. What is the effect of different initial investments on the time required to reach a certain amount?

#### Tip:
Remember, when interest is compounded more frequently, the investment grows faster due to the interest-on-interest effect!

How long will it take $15,000 to grow to $18,000 if the investment earns interest at the rate of 2%/year compounded monthly? (Round your answer to two decimal places.)

Learn how to calculate the time required for a $15,000 investment to grow to $18,000 at a 2% annual interest rate, compounded monthly. This solution explains key concepts in exponential growth, compound interest, and logarithmic equations, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation