To model the heights of the two trees over the years, we can create equations based on their growth rates.
-
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: \[ A(t) = 3 \times (1 + 0.20)^t = 3 \times (1.2)^t \]
-
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.