Question

Biologists are tracking the growth of a rabbit population. There were 45 rabbits initially and the population is quadrupling every year.

Which of the following equations can be used to find when the population of rabbits will be equal to 720?

Answer 1

To model the growth of the rabbit population, we start with the initial number of rabbits and the growth factor. The initial population is 45 rabbits, and the population quadruples every year. This can be expressed mathematically as:

\[
P(t) = P_0 \cdot r^t
\]

where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth factor, and
- \( t \) is the time in years.

In this case:
- \( P_0 = 45 \) (the initial population),
- \( r = 4 \) (the population quadruples, which means it is multiplied by 4 each year),
- We want to find when \( P(t) = 720 \).

Substituting the values into the equation:

\[
720 = 45 \cdot 4^t
\]

This equation can be rearranged to isolate \( 4^t \):

\[
4^t = \frac{720}{45}
\]

Calculating the right side:

\[
\frac{720}{45} = 16
\]

Thus, the equation simplifies to:

\[
4^t = 16
\]

Since \( 16 \) can be rewritten as \( 4^2 \), we can further express the equation as:

\[
4^t = 4^2
\]

From here, we can deduce that:

\[
t = 2
\]

Therefore, the equation that can be used to find when the population of rabbits will be equal to 720 is:

\[
720 = 45 \cdot 4^t
\]