Question

Gilberto plants two trees in his front yard. The apple tree is three feet tall and will grow 20 percent taller each year. The olive tree is two feet tall and will grow 30 percent taller each year. Create equations that model each tree’s height per year. How many years will it take for the trees to reach the same height?(1 point) Responses approximately 8 years approximately 8 years approximately 5 years approximately 5 years approximately 7 years approximately 7 years approximately 42 years approximately 42 years

Answer 1

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 \( t \):

1. Divide both sides by 2 to simplify:
\[ \frac{3}{2} \times (1.20)^t = (1.30)^t \]

2. Rearranging gives:
\[ (1.20)^t = \frac{2}{3} \times (1.30)^t \]

3. Taking logarithm on both sides:
\[ \log((1.20)^t) = \log\left(\frac{2}{3} \times (1.30)^t\right) \]

Using log properties:
\[ t \cdot \log(1.20) = \log\left(\frac{2}{3}\right) + t \cdot \log(1.30) \]

4. Rearranging to isolate \( t \):
\[ t \cdot \log(1.20) - t \cdot \log(1.30) = \log\left(\frac{2}{3}\right) \]

Factor out \( t \):
\[ t(\log(1.20) - \log(1.30)) = \log\left(\frac{2}{3}\right) \]

5. 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."