Let the height of the maple tree after \( t \) years be given by:
\[ M(t) = 2.5(1 + 0.17)^t = 2.5(1.17)^t \]
Let the height of the oak tree after \( t \) years be given by:
\[ O(t) = 5.6(1 + 0.08)^t = 5.6(1.08)^t \]
We need to find when these two heights are equal:
\[ 2.5(1.17)^t = 5.6(1.08)^t \]
Dividing both sides by 2.5 gives:
\[ (1.17)^t = \frac{5.6}{2.5}(1.08)^t \]
Calculating \( \frac{5.6}{2.5} \):
\[ \frac{5.6}{2.5} = 2.24 \]
The equation simplifies to:
\[ (1.17)^t = 2.24(1.08)^t \]
Rearranging gives us:
\[ \frac{(1.17)^t}{(1.08)^t} = 2.24 \]
This can be rewritten as:
\[ \left(\frac{1.17}{1.08}\right)^t = 2.24 \]
Calculating \( \frac{1.17}{1.08} \):
\[ \frac{1.17}{1.08} \approx 1.08333 \]
Now, rewriting the equation gives:
\[ 1.08333^t = 2.24 \]
Taking the logarithm of both sides:
\[ \log(1.08333^t) = \log(2.24) \]
Using the property of logarithms:
\[ t \cdot \log(1.08333) = \log(2.24) \]
Now solving for \( t \):
\[ t = \frac{\log(2.24)}{\log(1.08333)} \]
Calculating the values:
- \( \log(2.24) \approx 0.3508 \)
- \( \log(1.08333) \approx 0.0357 \)
Thus:
\[ t \approx \frac{0.3508}{0.0357} \approx 9.83 \]
Therefore, it will take approximately 10 years for the two trees to reach the same height. Thus, the correct response is:
Approximately 10 years.