Use the compound interest formula Upper A equals Upper P left parenthesis 1 plus r right parenthesis Superscript tA=P(1+r)t and the given information to solve for t.
Aequals=​$48 comma 00048,000​, Pequals=​$32 comma 00032,000​, requals=77​%

To solve for $ t $ using the compound interest formula $ A = P(1 + r)^t $, let's start by substituting the given values into the equation.

Given:
- $ A = 48,000 $
- $ P = 32,000 $
- $ r = 7\% = 0.07 $

The equation becomes:

\[
48,000 = 32,000(1 + 0.07)^t
\]

### Step 1: Divide both sides by 32,000

\[
\frac{48,000}{32,000} = (1.07)^t
\]

\[
1.5 = (1.07)^t
\]

### Step 2: Take the natural logarithm (ln) of both sides

\[
\ln(1.5) = \ln\left((1.07)^t\right)
\]

### Step 3: Apply the logarithm power rule

\[
\ln(1.5) = t \cdot \ln(1.07)
\]

### Step 4: Solve for $ t $

\[
t = \frac{\ln(1.5)}{\ln(1.07)}
\]

### Step 5: Calculate the value of $ t $

\[
t \approx \frac{0.4055}{0.0677} \approx 5.99
\]

So, $ t \approx 6 $ years.

### Conclusion

It will take approximately 6 years for the investment to grow from $32,000 to $48,000 at an interest rate of 7%.

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. How does changing the interest rate affect the time required for the investment to grow?
2. What is the formula for continuous compound interest?
3. What would be the final amount if the time was doubled?
4. How would the result change if the interest was compounded more frequently?
5. How can you solve for the interest rate $ r $ if you know the time $ t $ and the amounts $ A $ and $ P $?

**Tip:** Always check if the interest rate is compounded annually or more frequently; this can affect the outcome significantly.

Learn how to solve for time (t) using the compound interest formula A = P(1 + r)^t with a step-by-step solution. Given A = $48,000, P = $32,000, r = 7%. Find out how long it takes for an investment to grow from $32,000 to $48,000 at an interest rate of 7%.

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