The problem describes a population of cottontail rabbits that starts at 66 and quadruples every 16 years. The question asks how many years it takes for the population to reach 746.

1. Problem Description

The problem describes a population of cottontail rabbits that starts at 66 and quadruples every 16 years. The question asks how many years it takes for the population to reach

7

4

6.

2. Solution Steps

Let $P(t)$ be the population at time $t$, where $t$ is measured in years. The initial population is $P(0) = 66$.

Since the population quadruples every 16 years, we can write the population as an exponential function:

$P(t) = P(0) * 4^{t/16}$

We want to find the time $t$ when the population reaches

7

4

6. So, we set $P(t) = 746$:

$746 = 66 * 4^{t/16}$

Now, we solve for $t$:

$\frac{746}{66} = 4^{t/16}$

Taking the logarithm of both sides (using the natural logarithm, but any base will work):

$\ln\left(\frac{746}{66}\right) = \ln\left(4^{t/16}\right)$

Using the logarithm power rule:

$\ln\left(\frac{746}{66}\right) = \frac{t}{16} \ln(4)$

Now, solve for $t$:

$t = 16 * \frac{\ln\left(\frac{746}{66}\right)}{\ln(4)}$

$t = 16 * \frac{\ln(746/66)}{\ln(4)}$

$t = 16 * \frac{\ln(11.303)}{\ln(4)}$

$t = 16 * \frac{2.425}{1.386}$

$t = 16 * 1.75$

$t = 28.0018$

Rounding to the nearest hundredth, $t \approx 28.00$

3. Final Answer

28.00