To find out how many years it will take for the two trees to reach the same height, we can model the height of each tree with the following equations:
For the apple tree: \[ H_a(t) = 6 \times (1 + 0.05)^t = 6 \times (1.05)^t \]
For the cherry tree: \[ H_c(t) = 3 \times (1 + 0.16)^t = 3 \times (1.16)^t \]
We want to find \( t \) such that: \[ H_a(t) = H_c(t) \] This leads to the equation: \[ 6 \times (1.05)^t = 3 \times (1.16)^t \]
Now, we can simplify the equation: \[ 2 \times (1.05)^t = (1.16)^t \]
Divide both sides by \( (1.05)^t \): \[ 2 = \left( \frac{1.16}{1.05} \right)^t \]
Now, take the logarithm of both sides: \[ \log(2) = t \cdot \log\left(\frac{1.16}{1.05}\right) \]
Now we can solve for \( t \): \[ t = \frac{\log(2)}{\log\left(\frac{1.16}{1.05}\right)} \]
Now we calculate the values:
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Calculate \( \log(2) \): \[ \log(2) \approx 0.3010 \]
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Calculate \( \frac{1.16}{1.05} \): \[ \frac{1.16}{1.05} \approx 1.10476 \]
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Then, calculate \( \log(1.10476) \): \[ \log(1.10476) \approx 0.0427 \]
Now, plug these values into the equation for \( t \): \[ t \approx \frac{0.3010}{0.0427} \approx 7.05 \]
So, it will take approximately 7.0 years for the two trees to reach the same height.
Thus, the answer is Approximately 7.0 years.
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