Introduction:
The logistic growth model is a mathematical model that describes how a population grows over time. It takes into account both the growth rate and the carrying capacity of the environment. In this question, we are given a growth rate of 0.15 individuals per week and a carrying capacity of 400 individuals. We need to determine the maximum growth rate of the population.

Understanding the Logistic Growth Model:
The logistic growth model is based on the assumption that as the population size increases, the availability of resources becomes limited, leading to a decrease in the growth rate. Eventually, the population reaches a point where the growth rate becomes zero, called the carrying capacity.

The logistic growth model can be expressed mathematically as:

dN/dt = rN * (1 - N/K)

where:
dN/dt is the rate of change of population over time,
r is the intrinsic growth rate,
N is the population size, and
K is the carrying capacity.

Determining the Maximum Growth Rate:
In order to determine the maximum growth rate, we need to find the value of r that results in the maximum population growth. This occurs when the rate of change of population is at its peak.

To find the maximum growth rate, we can take the derivative of the logistic growth equation with respect to N and set it equal to zero:

dN/dt = rN * (1 - N/K)
dN/dt = r - (rN/K) - (rN^2 / K)

Setting dN/dt equal to zero and solving for N:

0 = r - (rN/K) - (rN^2 / K)
0 = r - (rN/K) - (rN^2 / K)

Simplifying the equation:

0 = r - 2rN/K - (rN^2 / K)

Since we are given the carrying capacity K = 400, we can substitute this value into the equation:

0 = r - 2rN/400 - (rN^2 / 400)

Simplifying further:

0 = r - (2rN + rN^2) / 400

We can rearrange the equation to find the values of N where the growth rate is zero:

r = (2rN + rN^2) / 400
400r = 2rN + rN^2
rN^2 + 2rN - 400r = 0

This is a quadratic equation in terms of N. Solving this equation will give us the values of N where the growth rate is zero. The maximum growth rate occurs when the population size is between these two values.

Conclusion:
To determine the maximum growth rate of the population, we need to solve the quadratic equation derived from the logistic growth model. The solution will give us the values of N where the growth rate is zero. The maximum growth rate occurs when the population size is between these two values. In this case, the correct answer is between 14.5 and 15.5.