17–18. {Use of Tech} Designing logistic functions Use the method of Example 1 to find a logistic function that describes the following populations. Graph the population function.

The population increases from 50 to 60 in the first month and eventually levels off at 150.

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Identify the general form of the logistic function: \(f(t) = \frac{L}{1 + Ce^{-kt}}\), where \(L\) is the carrying capacity (the population limit), \(C\) and \(k\) are constants to be determined, and \(t\) represents time in months.

From the problem, note that the population levels off at 150, so set \(L = 150\). Also, the initial population at \(t=0\) is 50, so use this to find \(C\) by substituting \(t=0\) and \(f(0) = 50\) into the logistic function.

Use the population at \(t=1\) month, which is 60, to create a second equation by substituting \(t=1\) and \(f(1) = 60\) into the logistic function. This will involve the unknown constant \(k\).

Solve the system of two equations obtained from steps 2 and 3 to find the values of \(C\) and \(k\). This may require algebraic manipulation and possibly taking natural logarithms.

Write the final logistic function with the determined constants \(L\), \(C\), and \(k\). Then, sketch or graph the function to visualize how the population grows from 50 towards the carrying capacity of 150 over time.

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Logistic Function Model

A logistic function models population growth that starts exponentially but slows as it approaches a maximum limit called the carrying capacity. It is typically expressed as P(t) = L / (1 + Ce^(-kt)), where L is the carrying capacity, and C and k are constants determined by initial conditions.

Parameter Determination Using Initial Conditions

To find the specific logistic function, you use given data points such as initial population values and growth at certain times. These conditions help solve for constants C and k by substituting values into the logistic equation and solving the resulting system of equations.

Graphing Logistic Functions

Graphing a logistic function involves plotting population over time, showing an S-shaped curve that starts near the initial population, rises rapidly, and then levels off at the carrying capacity. This visual helps interpret growth behavior and verify the model's accuracy.

17–18. {Use of Tech} Designing logistic functions Use the method of Example 1 to find a logistic function that describes the following populations. Graph the population function.The population increases from 50 to 60 in the first month and eventually levels off at 150.