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 solve for \( t \) in the exponential growth model \( A = 1,000,000e^{0.54t} \), where \( A \) is the value of the house at time \( t \), follow these steps:

### Given:
- The formula is \( A = 1,000,000e^{0.54t} \).
- \( A = 6,000,000 \) (the value of the house when we want to find \( t \)).

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

### Step 2: Divide both sides of the equation by 1,000,000 to isolate the exponential term.
\[
6 = e^{0.54t}
\]

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

### Step 4: Divide by 0.54 to find \( t \).
\[
t = \frac{\ln(6)}{0.54}
\]

### Step 5: Calculate the value of \( t \).
\[
t \approx \frac{1.79176}{0.54} \approx 3.3162
\]

### Final Answer:
\( t \approx 3.32 \) years.

So, the house would be worth P6,000,000 approximately 3.32 years after the initial purchase in 2016, which means around the year 2019.

Would you like more details on this process or have any questions?

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Here are 5 related questions you might find useful:

1. How would the value of the house change if the growth rate was different?
2. What would the value of the house be in 5 years?
3. How does exponential growth differ from linear growth in real estate?
4. Can the same method be applied to other investments with exponential growth?
5. How does changing the initial value affect the time \( t \)?

**Tip:** When solving exponential equations, logarithms are your best tool for isolating the variable of interest.

Learn how to calculate the time when a house reaches a certain value using the exponential growth model. Detailed steps include applying the exponential growth formula and natural logarithms to solve for the unknown time parameter. Suitable for high school students in Grades 10-12.

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