N(t) = 650

1+649 e−0.5 t
.

(You may wish to sketch a graph of N(t) to help you understand this situation and answer the questions below.)

(a) How many people will have heard the rumor after 2 hours?

After 12 hours?

(b) When will half the people have heard the rumor? t =

(c) When is the rumor spreading fastest? t =

To answer these questions, we need to plug in the given values of t into the equation N(t) = 650 / (1 + 649e^(-0.5t)) and solve for N(t).

(a) To find the number of people who will have heard the rumor after 2 hours, we substitute t = 2 into the equation:
N(2) = 650 / (1 + 649e^(-0.5 * 2))

To find the number of people who will have heard the rumor after 12 hours, we substitute t = 12 into the equation:
N(12) = 650 / (1 + 649e^(-0.5 * 12))

(b) To find when half the people will have heard the rumor, we set N(t) = 325 and solve for t:
325 = 650 / (1 + 649e^(-0.5t))

To solve this equation, we can multiply both sides by the denominator and rearrange the equation to isolate the exponential term:
(1 + 649e^(-0.5t)) = 650 / 325
1 + 649e^(-0.5t) = 2
649e^(-0.5t) = 1

Then, we can solve for e^(-0.5t) by dividing both sides by 649:
e^(-0.5t) = 1 / 649

To solve for t, we take the natural logarithm (ln) of both sides:
ln(e^(-0.5t)) = ln(1/649)
-0.5t = ln(1/649)

Finally, we solve for t by dividing both sides by -0.5:
t = (ln(1/649)) / -0.5

(c) To find when the rumor is spreading fastest, we need to find the maximum value of N(t). We can achieve this by finding the derivative of N(t) with respect to t and setting it equal to zero.

First, we find the derivative of N(t) with respect to t:
N'(t) = (dN(t) / dt) = (-650 * (-0.5 * 649 * e^(-0.5t))) / (1 + 649e^(-0.5t))^2

To find the critical points, we set N'(t) equal to zero and solve for t:
0 = (-650 * (-0.5 * 649 * e^(-0.5t))) / (1 + 649e^(-0.5t))^2

Since the denominator cannot be zero, we set the numerator equal to zero:
0 = -650 * (-0.5 * 649 * e^(-0.5t))

Next, we solve for e^(-0.5t):
0 = 0.5 * 649 * e^(-0.5t)

To solve this equation, we divide both sides by 0.5 * 649:
0 = e^(-0.5t)

Then, we take the natural logarithm (ln) of both sides:
ln(e^(-0.5t)) = ln(0)

Therefore, there is no solution for t in this case.

Overall, plug in the values into the equation and perform the necessary calculations to find the answers to the questions (a), (b), and (c).

Do you mean to divide?

N(t) = 650/(1+649e^-.5t)

What's the trouble? Straightforward evaluation of N or solving for t. That's just algebra.

If I have the function right, its graphs is

http://www.wolframalpha.com/input/?i=650%2F%281%2B649e^%28-.5t%29%29