Question

Gilberto plants two trees in his front yard. The apple tree is three feet tall and will grow 20 percent taller each year. The olive tree is two feet tall and will grow 30 percent taller each year.
Create equations that model each tree's height per year. How many years will it take for the trees to reach the same height?

Answer 1

To model the heights of the two trees over the years, we can create equations based on their growth rates.

1. **Apple Tree:**
- Initial height: 3 feet
- Growth rate: 20% per year
- After \( t \) years, the height of the apple tree can be modeled by the equation:
\[
A(t) = 3 \times (1 + 0.20)^t = 3 \times (1.2)^t
\]

2. **Olive Tree:**
- Initial height: 2 feet
- Growth rate: 30% per year
- After \( t \) years, the height of the olive tree can be modeled by the equation:
\[
O(t) = 2 \times (1 + 0.30)^t = 2 \times (1.3)^t
\]

Next, we want to find the value of \( t \) where the heights of the two trees are equal:
\[
A(t) = O(t)
\]

Setting the two equations equal to each other:
\[
3 \times (1.2)^t = 2 \times (1.3)^t
\]

To solve for \( t \), we can rearrange the equation:
\[
\frac{(1.2)^t}{(1.3)^t} = \frac{2}{3}
\]

This simplifies to:
\[
\left(\frac{1.2}{1.3}\right)^t = \frac{2}{3}
\]

Taking the natural logarithm of both sides:
\[
t \cdot \ln\left(\frac{1.2}{1.3}\right) = \ln\left(\frac{2}{3}\right)
\]

Now, we can solve for \( t \):
\[
t = \frac{\ln\left(\frac{2}{3}\right)}{\ln\left(\frac{1.2}{1.3}\right)}
\]

Calculating the values:
- \(\ln\left(\frac{2}{3}\right) \approx -0.4055\)
- \(\ln\left(\frac{1.2}{1.3}\right) \approx -0.0834\)

Now substituting the values into the equation:
\[
t \approx \frac{-0.4055}{-0.0834} \approx 4.86
\]

Since \( t \) must be a whole number, we round up to the next whole number:
\[
t \approx 5 \text{ years}
\]

So, it will take approximately **5 years** for both trees to reach the same height.