To model the height of each tree over the years, we'll use the formula for exponential growth:
\[ H(t) = H_0 \times (1 + r)^t \]
where:
- \( H(t) \) is the height at time \( t \),
- \( H_0 \) is the initial height,
- \( r \) is the growth rate,
- \( t \) is the number of years.
For the Apple Tree:
- Initial height (\( H_0 \)) = 3 ft
- Growth rate (\( r \)) = 20% = 0.20
The equation for the height of the apple tree after \( t \) years is:
\[ H_a(t) = 3 \times (1 + 0.20)^t = 3 \times (1.20)^t \]
For the Olive Tree:
- Initial height (\( H_0 \)) = 2 ft
- Growth rate (\( r \)) = 30% = 0.30
The equation for the height of the olive tree after \( t \) years is:
\[ H_o(t) = 2 \times (1 + 0.30)^t = 2 \times (1.30)^t \]
Setting the Heights Equal
To find out when the two trees will reach the same height, we need to solve the equation:
\[ 3 \times (1.20)^t = 2 \times (1.30)^t \]
Simplifying the Equation
-
Divide both sides by 2:
\[ \frac{3}{2} \times (1.20)^t = (1.30)^t \]
-
To isolate \( t \), we can take logarithms of both sides:
\[ \log\left(\frac{3}{2}\right) + t \log(1.20) = t \log(1.30) \]
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Rearranging gives:
\[ \log\left(\frac{3}{2}\right) = t \left( \log(1.30) - \log(1.20) \right) \]
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Solving for \( t \):
\[ t = \frac{\log\left(\frac{3}{2}\right)}{\log(1.30) - \log(1.20)} \]
Calculating \( t \)
Now we can calculate \( t \) using approximate logarithm values:
- \( \log(1.30) \approx 0.113943 \)
- \( \log(1.20) \approx 0.079181 \)
- \( \log\left(\frac{3}{2}\right) \approx \log(1.5) \approx 0.176091 \)
Putting these into our equation:
\[ t \approx \frac{0.176091}{0.113943 - 0.079181} \approx \frac{0.176091}{0.034762} \approx 5.06 \]
Conclusion
It will take approximately \( 5 \) years for the two trees to reach the same height.
The answer is approximately 5 years (option 3).