Question

Xochitl planted two saplings in her yard. The maple tree is 2.5 feet tall and will grow at a rate of 17% each year. The oak tree is 5.6 feet tall and grows at a rate of 8% each year. How many years will it take the two trees to reach the same height?

A. 2.5 years
B. They will never be the same height
C. 10 years
D. 34 years

Answer 1

To find out when the heights of the two trees will be the same, we can use the formula for exponential growth. The formula for the height of each tree after \( t \) years is:

1. For the maple tree:
\[
H_m(t) = 2.5 \times (1 + 0.17)^t = 2.5 \times (1.17)^t
\]

2. For the oak tree:
\[
H_o(t) = 5.6 \times (1 + 0.08)^t = 5.6 \times (1.08)^t
\]

We need to find \( t \) such that:
\[
2.5 \times (1.17)^t = 5.6 \times (1.08)^t
\]

Dividing both sides by 2.5:
\[
(1.17)^t = \frac{5.6}{2.5} \times (1.08)^t
\]
\[
(1.17)^t = 2.24 \times (1.08)^t
\]

Now, we can rearrange this to take the logarithm of both sides:
\[
\left(\frac{1.17}{1.08}\right)^t = 2.24
\]

Taking the logarithm, we get:
\[
t \cdot \log\left(\frac{1.17}{1.08}\right) = \log(2.24)
\]
\[
t = \frac{\log(2.24)}{\log\left(\frac{1.17}{1.08}\right)}
\]

Now, let's calculate \( \log(2.24) \) and \( \log\left(\frac{1.17}{1.08}\right) \):
- \( \log(2.24) \approx 0.352 \)
- \( \log\left(\frac{1.17}{1.08}\right) = \log(1.17) - \log(1.08) \approx 0.071 - 0.033 = 0.038 \)

Now, substituting the values:
\[
t \approx \frac{0.352}{0.038} \approx 9.26
\]

Rounding to the nearest whole number, it will take approximately 9 years for the two trees to reach the same height.

Given the options:
A. 2.5 years
B. They will never be the same height
C. 10 years
D. 34 years

The closest and most appropriate answer is **C. 10 years**.