Question

Naya planted two fruit trees in her yard. The apple tree is 6 feet tall and will grow at a rate of 5% each year. The cherry tree is 3 feet tall and grows at a rate of 16% each year. How many years will it take the two trees to reach the same height?

A. 2.1 years
B. 4.3 years
C. 7.0 years
D. 8.4 years

Answer 1

To solve for the number of years it will take for the apple tree and cherry tree to reach the same height, we can use the formula for exponential growth:

For the apple tree:
\[
h_a = h_{a_0} \times (1 + r_a)^t
\]
where:
- \( h_{a_0} = 6 \) feet (initial height)
- \( r_a = 0.05 \) (growth rate of 5%)
- \( t \) = number of years

This simplifies to:
\[
h_a = 6 \times (1.05)^t
\]

For the cherry tree:
\[
h_c = h_{c_0} \times (1 + r_c)^t
\]
where:
- \( h_{c_0} = 3 \) feet (initial height)
- \( r_c = 0.16 \) (growth rate of 16%)

This simplifies to:
\[
h_c = 3 \times (1.16)^t
\]

We want to find the time \( t \) when \( h_a = h_c \):
\[
6 \times (1.05)^t = 3 \times (1.16)^t
\]

Dividing both sides by 3 gives:
\[
2 \times (1.05)^t = (1.16)^t
\]

Next, we can isolate the exponential terms:
\[
\frac{(1.16)^t}{(1.05)^t} = 2
\]
This can be rewritten as:
\[
\left(\frac{1.16}{1.05}\right)^t = 2
\]

To solve for \( t \), we take the logarithm of both sides:
\[
t \cdot \log\left(\frac{1.16}{1.05}\right) = \log(2)
\]

Now, calculate \( \frac{1.16}{1.05} \):
\[
\frac{1.16}{1.05} \approx 1.10476
\]

Now we find the logs:
\[
\log(1.10476) \approx 0.04245
\]
\[
\log(2) \approx 0.30103
\]

Now substituting back into our equation for \( t \):
\[
t = \frac{\log(2)}{\log\left(\frac{1.16}{1.05}\right)} \approx \frac{0.30103}{0.04245} \approx 7.09
\]

Thus, the number of years for the two trees to reach the same height is approximately **7.1 years**.

The closest answer choice is:
\[
\text{C. 7.0 years}
\]