Four hundred students attended the dedication ceremony of a new building on a college campus. The president of the traditionally female college announced a new expansion program which included plans to make the college coeducational. The number of students who learned of the new program t hr later is given by the function below.

f(t) = 4000/(1+B e^(-k t))
If 800 students on campus had heard about the new program 2 hr after the ceremony, how many students had heard about the policy after 4 hr?
students

How fast was the news spreading after 4 hr?
students/hr

Any help?

To find out how many students had heard about the policy after 4 hours, we can substitute the values given into the function f(t) and solve for the unknown variable.

Given values:
Initial number of students: f(0) = 400
Number of students after 2 hours: f(2) = 800
Time after 4 hours: t = 4

Substituting these values into the function, we get:
800 = 4000 / (1 + B * e^(-4k))

To solve for the unknown variables B and k, we need another equation. We can use the value of the function after 2 hours:

800 = 4000 / (1 + B * e^(-2k))

Now we have a system of two equations with two variables. We can solve this system using algebraic techniques such as substitution or elimination.

Divide the two equations to eliminate the constant term:
(4000 / (1 + B * e^(-4k))) / (4000 / (1 + B * e^(-2k))) = 1

Simplifying, we get:
(1 + B * e^(-2k)) / (1 + B * e^(-4k)) = 1

Cross-multiplying, we have:
1 + B * e^(-2k) = 1 + B * e^(-4k)

Simplifying further, we get:
B * e^(-2k) = B * e^(-4k)

Dividing by B, we have:
e^(-2k) = e^(-4k)

Taking the natural log of both sides, we get:
-2k = -4k

Solving for k, we find:
2k = 4k
k = 0

Now that we have found k, we can substitute it back into one of the equations to solve for B. Let's use the equation f(2) = 800:

800 = 4000 / (1 + B * e^(-2k))
800 = 4000 / (1 + B * e^(0))
800 = 4000 / (1 + B)

Multiplying both sides by (1 + B), we get:
800(1 + B) = 4000

Distributing, we have:
800 + 800B = 4000

Subtracting 800 from both sides, we get:
800B = 3200

Dividing by 800, we find:
B = 4

Now we finally have the values for both B and k, we can substitute these values back into the original function and solve for f(4) to find out how many students had heard about the policy after 4 hours:

f(t) = 4000 / (1 + B e^(-k t))
f(4) = 4000 / (1 + 4 * e^(-0 * 4))
f(4) = 4000 / (1 + 4 * e^0)
f(4) = 4000 / (1 + 4 * 1) [since e^0 = 1]
f(4) = 4000 / (1 + 4)
f(4) = 4000 / 5
f(4) = 800

Therefore, after 4 hours, 800 students had heard about the policy.

To find out how fast the news was spreading after 4 hours, we can find the derivative of the function f(t). The derivative gives us the rate of change of the function with respect to time. In other words, it tells us how fast the number of students who heard about the policy is increasing or decreasing at any given time.

Taking the derivative of the function f(t), we get:
f'(t) = -4000 * (-k) * e^(-k t) / (1 + B e^(-k t))^2

Substituting the values of B and k we found earlier, we get:
f'(t) = -4000 * 0 * e^(0) / (1 + 4 * e^(0))^2
f'(t) = 0 / (1 + 4)^2
f'(t) = 0 / 25
f'(t) = 0

Therefore, the news was not spreading or changing at all after 4 hours. The rate of change was 0 students per hour.