Question

Many biological populations can be modeled by f(t) = f(0) eᶜ ᶜᵒˢ ᵗ, where f(0) is the size of the population when t = 0. Suppose that f(0) = 1100 and c=4. Find the maximum and minimum values of f(t) and the values of t where they occur.

Find the maximum value(s) of f(t) and the value(s) of t where they occur. Select the correct answers below and, if necessary, fill in the answer boxes to complete your choice.

A. The maximum value of f(t) is___, which on the interval 0≤t<2t occurs at t= ___(Type exact answers, using a as needed. Use a comma to separate answers as needed.)

Answer 1

To find the maximum and minimum values of the given population model, substitute the given values into the equation. The maximum value occurs at t = 0, and the minimum value occurs at t = π.

Step-by-step explanation:

To find the maximum and minimum values of f(t) and the values of t where they occur for the given population model f(t) = f(0) * e^(c*cos(t)), we can substitute the given values of f(0) = 1100 and c = 4 into the equation.

1. Calculate the maximum and minimum values of f(t):

f(t) = 1100 * e^(4*cos(t))

The maximum value will occur when cos(t) = 1, which is at t = 0.

The minimum value will occur when cos(t) = -1, which is at t = π.

Substituting these values into the equation, we find:

Maximum value: f(0) = 1100 * e^(4*cos(0)) = 1100 * e^4

Minimum value: f(π) = 1100 * e^(4*cos(π)) = 1100 * e^(-4)

2. Calculate the values of t where the maximum and minimum values occur:

The maximum value occurs at t = 0. As for the minimum value, it occurs at t = π.