19. Under ideal conditions, a population of rabbits has an exponential growth rate of $$ 11.7 \% $$ per day. Consider an initial population of 100 rabbits. How long before we have 500?

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To determine how long it will take for a population of rabbits to grow from 100 to 500 under an exponential growth rate of 11.7% per day, we can use the formula for exponential growth:

$$
P(t) = P_0 e^{rt}
$$

where:
- $$ P(t) $$ is the population at time $$ t $$,
- $$ P_0 $$ is the initial population,
- $$ r $$ is the growth rate (as a decimal),
- $$ t $$ is the time in days,
- $$ e $$ is the base of the natural logarithm.

Given:
- $$ P_0 = 100 $$
- $$ P(t) = 500 $$
- $$ r = 11.7\% = 0.117 $$

We need to solve for $$ t $$ when $$ P(t) = 500 $$:

$$
500 = 100 e^{0.117t}
$$

First, divide both sides by 100:

$$
5 = e^{0.117t}
$$

Next, take the natural logarithm of both sides:

$$
\ln(5) = \ln(e^{0.117t})
$$

Using the property of logarithms that $$ \ln(e^x) = x $$:

$$
\ln(5) = 0.117t
$$

Now, solve for $$ t $$:

$$
t = \frac{\ln(5)}{0.117}
$$

Calculating $$ \ln(5) $$:

$$
\ln(5) \approx 1.60944
$$

Now substitute this value into the equation for $$ t $$:

$$
t \approx \frac{1.60944}{0.117} \approx 13.74
$$

Thus, it will take approximately **13.74 days** for the rabbit population to grow from 100 to 500 under ideal conditions.
It will take approximately 13.74 days for the rabbit population to grow from 100 to 500.

Under ideal conditions, a population of rabbits has an exponential growth
rate of per day. Consider an initial population of 100 rabbits. How long
before we have 500?

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Under ideal conditions, a population of rabbits has an exponential growth rate of per day. Consider an initial population of 100 rabbits. How long before we have 500?