We are given that 53 white-tail deer were introduced into a state park. The population quadruples every 14 years. We need to find the number of years it takes for the population to reach 630.

1. Problem Description

We are given that 53 white-tail deer were introduced into a state park. The population quadruples every 14 years. We need to find the number of years it takes for the population to reach

6

3

0.

2. Solution Steps

Let $P(t)$ be the population at time $t$ years.

The initial population is $P(0) = 53$.

The population quadruples every 14 years, so $P(14) = 4 \times 53 = 212$.

The general formula for exponential growth is:

$P(t) = P(0) \times (growth\ rate)^{(t/time\ period)}$

In our case, the growth rate is 4, and the time period is 14 years.

So, $P(t) = 53 \times 4^{(t/14)}$.

We want to find the time $t$ when $P(t) = 630$.

$630 = 53 \times 4^{(t/14)}$

Divide both sides by 53:

$\frac{630}{53} = 4^{(t/14)}$

Take the natural logarithm of both sides:

$ln(\frac{630}{53}) = ln(4^{(t/14)})$

Using the property of logarithms, $ln(a^b) = b \times ln(a)$:

$ln(\frac{630}{53}) = \frac{t}{14} \times ln(4)$

Now, solve for $t$:

$t = 14 \times \frac{ln(\frac{630}{53})}{ln(4)}$

$t = 14 \times \frac{ln(630/53)}{ln(4)}$

$t \approx 14 \times \frac{2.4655}{1.3863}$

$t \approx 14 \times 1.7785$

$t \approx 24.899$

Rounding to the nearest hundredth of a year:

$t \approx 24.90$

3. Final Answer

24.90