To model the height of each tree over the years, we can create exponential growth equations for both the apple tree and the olive tree.
Apple tree:
- Initial height: 3 feet
- Growth rate: 20% (which means the height will be multiplied by 1.20 each year)
The equation for the height of the apple tree after \( t \) years is: \[ H_a(t) = 3 \times (1.20)^t \]
Olive tree:
- Initial height: 2 feet
- Growth rate: 30% (which means the height will be multiplied by 1.30 each year)
The equation for the height of the olive tree after \( t \) years is: \[ H_o(t) = 2 \times (1.30)^t \]
To find out when the two trees will be the same height, we set the two equations equal to each other: \[ 3 \times (1.20)^t = 2 \times (1.30)^t \]
Now we solve for \( t \):
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Divide both sides by 2 to simplify: \[ \frac{3}{2} \times (1.20)^t = (1.30)^t \]
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Rearranging gives: \[ (1.20)^t = \frac{2}{3} \times (1.30)^t \]
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Taking logarithm on both sides: \[ \log((1.20)^t) = \log\left(\frac{2}{3} \times (1.30)^t\right) \]
Using log properties: \[ t \cdot \log(1.20) = \log\left(\frac{2}{3}\right) + t \cdot \log(1.30) \]
- Rearranging to isolate \( t \): \[ t \cdot \log(1.20) - t \cdot \log(1.30) = \log\left(\frac{2}{3}\right) \]
Factor out \( t \): \[ t(\log(1.20) - \log(1.30)) = \log\left(\frac{2}{3}\right) \]
- Solving for \( t \): \[ t = \frac{\log\left(\frac{2}{3}\right)}{\log(1.20) - \log(1.30)} \]
Now we can use a calculator to evaluate:
- \( \log\left(\frac{2}{3}\right) \approx -0.1761 \)
- \( \log(1.20) \approx 0.0792 \)
- \( \log(1.30) \approx 0.1139 \)
Now plugging in the values: \[ t = \frac{-0.1761}{0.0792 - 0.1139} = \frac{-0.1761}{-0.0347} \approx 5.07 \]
Thus, it will take approximately 5 years for the trees to reach the same height. The correct answer is "approximately 5 years."