Questions: An initial population of 30 deer is moved to an uninhabited island. The population is modeled by P(t) = 1100 / (1 + 41 e^(-0.065 t)) where t is the months since the deer were first introduced. a. Find the point of inflection. Use the fact that the output value at the point of inflection of a logistic function is half the limiting population. (Round each coordinate to two decimal places.) b. Find the rate of change at the point of inflection and include units. (Round to two decimal places.) c. What can you conclude about the rate of growth at the point of inflection? The rate of growth is . d. What is the limiting population of the deer? deer

An initial population of 30 deer is moved to an uninhabited island. The population is modeled by
P(t) = 1100 / (1 + 41 e^(-0.065 t))
where t is the months since the deer were first introduced.
a. Find the point of inflection. Use the fact that the output value at the point of inflection of a logistic function is half the limiting population. (Round each coordinate to two decimal places.)

b. Find the rate of change at the point of inflection and include units. (Round to two decimal places.)

c. What can you conclude about the rate of growth at the point of inflection?

The rate of growth is .

d. What is the limiting population of the deer?

deer

Transcript text: An initial population of 30 deer is moved to an uninhabited island. The population is modeled by \[ P(t)=\frac{1100}{1+41 e^{-0.065 t}} \] where $t$ is the months since the deer were first introduced. a. Find the point of inflection. Use the fact that the output value at the point of inflection of a logistic function is half the limiting population. (Round each coordinate to two decimal places.) $\square$ $\square$ b. Find the rate of change at the point of inflection and include units. (Round to two decimal places.) $\square$ c. What can you conclude about the rate of growth at the point of inflection? The rate of growth is $\square$ . d. What is the limiting population of the deer? $\square$ deer

Solution

Solution Steps

To solve the given problem, we need to follow these steps:

a. Find the point of inflection:

The point of inflection for a logistic function occurs where the population is half of the limiting population.
Calculate the limiting population by evaluating the function as $ t $ approaches infinity.
Determine the time $ t $ when the population is half of the limiting population.

b. Find the rate of change at the point of inflection:

Compute the derivative of the population function $ P(t) $.
Evaluate the derivative at the point of inflection to find the rate of change.

c. Conclusion about the rate of growth at the point of inflection:

The rate of growth at the point of inflection is the maximum rate of growth for the logistic function.

Step 1: Limiting Population

The limiting population $ L $ of the deer is given by evaluating the function $ P(t) $ as $ t $ approaches infinity. Thus, we have: \[ L = 1100 \]

Step 2: Point of Inflection

The point of inflection occurs when the population is half of the limiting population. Therefore, we calculate: \[ \text{Half of } L = \frac{1100}{2} = 550 \] The time $ t $ at which this occurs is approximately: \[ t \approx 57.13 \] Thus, the coordinates of the point of inflection are: \[ (t, P(t)) \approx (57.13, 550.00) \]

Step 3: Rate of Change at the Point of Inflection

To find the rate of change at the point of inflection, we evaluate the derivative $ P'(t) $ at $ t \approx 57.13 $: \[ P'(t) \approx 17.87 \text{ deer per month} \]

Step 4: Conclusion about the Rate of Growth

At the point of inflection, the rate of growth is the maximum rate of growth for the logistic function.