To model the height of each tree over the years, we can use the formula for exponential growth:
\[ H(t) = H_0 \times (1 + r)^t \]
where:
- \( H(t) \) is the height at year \( t \),
- \( H_0 \) is the initial height,
- \( r \) is the growth rate (in decimal form),
- \( t \) is the number of years.
For the Apple Tree:
- Initial height \( H_0 = 3 \) feet
- Growth rate \( r = 0.20 \)
The equation for the apple tree's height will be: \[ H_a(t) = 3 \times (1 + 0.20)^t = 3 \times (1.20)^t \]
For the Olive Tree:
- Initial height \( H_0 = 2 \) feet
- Growth rate \( r = 0.30 \)
The equation for the olive tree's height will be: \[ H_o(t) = 2 \times (1 + 0.30)^t = 2 \times (1.30)^t \]
Finding when the heights are equal:
To find out when the trees reach the same height, we need to set their equations equal to each other: \[ 3 \times (1.20)^t = 2 \times (1.30)^t \]
Now, we can simplify and solve for \( t \):
- Divide both sides by 2: \[ 1.5 \times (1.20)^t = (1.30)^t \]
- Divide both sides by \( (1.20)^t \): \[ 1.5 = \left(\frac{1.30}{1.20}\right)^t \]
Now we will calculate \( \frac{1.30}{1.20} \): \[ \frac{1.30}{1.20} \approx 1.0833 \]
So we rewrite the equation: \[ 1.5 = (1.0833)^t \]
Next, we take the logarithm of both sides: \[ \log(1.5) = t \cdot \log(1.0833) \] \[ t = \frac{\log(1.5)}{\log(1.0833)} \]
Calculating the logarithms:
- \( \log(1.5) \approx 0.1761 \)
- \( \log(1.0833) \approx 0.0351 \)
Now, calculating \( t \): \[ t \approx \frac{0.1761}{0.0351} \approx 5.02 \]
Final Answer:
Thus, it will take approximately 5 years for the trees to reach the same height.