The growth rate of a population of rabbits in a forest preserve is modeled by the logistic equation dp/dt =0.005p(300-0) . What is the rate of change in the rabbit population when it is growing the fastest?

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To determine the rate of change in the rabbit population when it is growing the fastest, we need to find the maximum point on the growth curve. In this case, the growth rate is modeled by the logistic equation:

dp/dt = 0.005p(300 - p)

To find the maximum point, we need to find the value of p that maximizes the growth rate dp/dt. We can do this by finding the derivative of dp/dt with respect to p and setting it equal to zero.

Let's differentiate dp/dt with respect to p:

d(dp/dt)/dp = d/dp (0.005p(300 - p))
= 0.005(300 - 2p)

Now, set this derivative equal to zero and solve for p:

0.005(300 - 2p) = 0

Simplifying the equation:

300 - 2p = 0

2p = 300

p = 150

So, when the rabbit population is 150, it is growing the fastest.

Now, to find the rate of change in the rabbit population at this point, substitute p = 150 into the growth rate equation:

dp/dt = 0.005p(300 - p)
= 0.005(150)(300 - 150)
= 0.005(150)(150)
= 11.25

Therefore, the rate of change in the rabbit population when it is growing the fastest is 11.25.