2. A house is purchased for P1,500,000 in 2016. The value of the house is given by the exponential growth model A = 1,000,000e 0.54t. Find t when the house would be worth P6,000,000.

To find the time \( t \) when the house would be worth \( P6,000,000 \), we can use the given exponential growth model:

\[
A = 1,000,000e^{0.54t}
\]

where:
- \( A \) is the value of the house at time \( t \),
- \( t \) is the time in years since 2016.

We are asked to find \( t \) when \( A = 6,000,000 \).

### Step 1: Substitute the values into the equation
\[
6,000,000 = 1,000,000e^{0.54t}
\]

### Step 2: Divide both sides by 1,000,000
\[
6 = e^{0.54t}
\]

### Step 3: Take the natural logarithm (ln) of both sides to solve for \( t \)
\[
\ln(6) = \ln(e^{0.54t})
\]

Since \( \ln(e^x) = x \), this simplifies to:
\[
\ln(6) = 0.54t
\]

### Step 4: Solve for \( t \)
\[
t = \frac{\ln(6)}{0.54}
\]

### Step 5: Calculate the value of \( t \)
\[
t \approx \frac{1.7918}{0.54} \approx 3.3174 \text{ years}
\]

### Conclusion:
The house would be worth \( P6,000,000 \) approximately 3.32 years after 2016, which means around mid-2019.

Would you like more details, or do you have any questions?

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Here are 5 related questions you might find interesting:

1. How would the model change if the growth rate were different?
2. What would the value of the house be after 10 years?
3. How can we interpret the constant 0.54 in the context of this problem?
4. What if the initial value of the house were different—how would that affect the time \( t \)?
5. How does this model compare to a linear growth model?

**Tip:** When solving exponential equations, taking the natural logarithm is a powerful tool for isolating the variable.

Learn how to calculate the time when a house would be worth P6,000,000 using the exponential growth model. This problem involves understanding exponential growth, using natural logarithms, and applying them to determine time-based value changes of assets.

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