okay sorry but this is the last problem that I don't understand how to start...

A population of prairie dogs grows exponentially. The colony begins with 32 prairie dogs; 3 years later there are 200 prairie dogs. Give a formula for population, P, as a function of time.

help would be appreciated, thank you :)

p = 32 e^(kt) , where k is a constant

when t = 3, p = 200

32 e^(3k) = 200
e^(3k) = 6.25
3k = ln 6.25
k = appr .61086

p = 32 e^(.61086 t)

can you please tell me how did you know which formula to use?

exponential growth is always of the form

p = a*e^(kt)

that's why it's called exponential...

The starting amount (at t=0) is a, since e^0 = 1.

To find a formula for the population, P, as a function of time, we need to first understand the concept of exponential growth and then work through the problem step by step.

Exponential growth occurs when a quantity or population increases by a fixed percentage over a fixed interval of time. In this case, we know that the population of prairie dogs grows exponentially.

Let's break down the information given in the problem:

- The colony begins with 32 prairie dogs.
- 3 years later, there are 200 prairie dogs.

We can use this information to find the growth rate per year and then use it to create a formula.

Step 1: Determine the growth rate per year
To find the growth rate per year, we can use the formula for exponential growth, which is of the form P = P₀ * e^(rt), where P is the final population, P₀ is the initial population, r is the growth rate per time period, and t is the time period.

Since we are given two points in time:

- At time t₁ = 0 (the initial population), P = 32.
- At time t₂ = 3 years later, P = 200.

We can substitute these values into the equation to solve for the growth rate, r.
Using P = P₀ * e^(rt) with t = 3:

200 = 32 * e^(3r)

Divide both sides of the equation by 32 to isolate e^(3r):

200/32 = e^(3r)

6.25 = e^(3r)

Take the natural logarithm (ln) of both sides to solve for 3r:

ln(6.25) = ln(e^(3r))

ln(6.25) = 3r * ln(e)

ln(6.25) = 3r * 1 (ln(e) = 1)

So, 3r = ln(6.25).

Divide both sides by 3 to find the growth rate per year, r:

r = ln(6.25) / 3.

Step 2: Construct the formula for population, P, as a function of time
Using the growth rate per year, r, we can now construct the formula for the population, P, as a function of time, t.

P = P₀ * e^(rt)

Substituting the given initial population of 32:

P = 32 * e^((ln(6.25)/3) * t)

Simplifying further:

P = 32 * e^(ln(6.25t) / 3)

Since e^(ln(x)) equals x, we have:

P = 32 * (6.25t)^(1/3)

Therefore, the formula for population, P, as a function of time, t, is:

P = 32 * (6.25t)^(1/3)

This formula will allow you to calculate the population of prairie dogs at any given time, t, using the initial population of 32 and the growth rate per year, which we found to be ln(6.25) / 3.

Remember to substitute the desired value for t into the formula to get the specific population at that time.