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) approximately 42 years approximately 8 years approximately 5 years approximately 7 years

Answer 1

To model the height of each tree over the years, we'll create equations based on the growth rates provided.

1. **Apple Tree**:
- Initial height: 3 feet
- Growth rate: 20% increase per year
- Equation:
\[
A(t) = 3 \times (1 + 0.20)^t = 3 \times (1.20)^t
\]
where \( A(t) \) is the height of the apple tree after \( t \) years.

2. **Olive Tree**:
- Initial height: 2 feet
- Growth rate: 30% increase per year
- Equation:
\[
O(t) = 2 \times (1 + 0.30)^t = 2 \times (1.30)^t
\]
where \( O(t) \) is the height of the olive tree after \( t \) years.

Next, to find out after how many years the trees will be the same height, we need to set their height equations equal to each other:
\[
3 \times (1.20)^t = 2 \times (1.30)^t
\]

To solve for \( t \), we can rearrange the equation:
\[
\frac{3}{2} = \frac{(1.30)^t}{(1.20)^t}
\]
\[
\frac{3}{2} = \left(\frac{1.30}{1.20}\right)^t
\]
Next, compute \( \frac{1.30}{1.20} \):
\[
\frac{1.30}{1.20} \approx 1.0833
\]

So we can rewrite the equation as:
\[
\frac{3}{2} = (1.0833)^t
\]

Now, taking the logarithm of both sides, we have:
\[
\log(\frac{3}{2}) = t \cdot \log(1.0833)
\]
To isolate \( t \), we can express it as:
\[
t = \frac{\log(\frac{3}{2})}{\log(1.0833)}
\]

Now we can calculate it:
1. Calculate \( \log(\frac{3}{2}) \):
\[
\log(1.5) \approx 0.1761
\]

2. Calculate \( \log(1.0833) \):
\[
\log(1.0833) \approx 0.0352
\]

Now plug in these values:
\[
t \approx \frac{0.1761}{0.0352} \approx 5
\]

Thus, it will take approximately **5 years** for the two trees to reach the same height. Therefore, the answer is **approximately 5 years**.