Math Help - Help with Derivatives in Word Problems

1. Help with Derivatives in Word Problems

A spread of a rumor of free KFC chicken in the school lobby is modeled by the equation P(t)= (224)/(1+e^(5-t)), where P(t) is the total number of students who have heard t minutes after the deep-fried goodness has arrived.

a. Estimate the initial number of students who first heard about the giveaway of these finger-lickin' delights.

b. How fast is the rumor of sweet,tangy,belly-filling joy spreading after 4 minutes?

c. When is this rumor spreading at its maximum rate?

d. What is this rate?

2. Originally Posted by Chris22
A spread of a rumor of free KFC chicken in the school lobby is modeled by the equation P(t)= (224)/(1+e^(5-t)), where P(t) is the total number of students who have heard t minutes after the deep-fried goodness has arrived.

a. Estimate the initial number of students who first heard about the giveaway of these finger-lickin' delights.

P(0)

b. How fast is the rumor of sweet,tangy,belly-filling joy spreading after 4 minutes?

P'(4)

c. When is this rumor spreading at its maximum rate?

when P''(t) = 0

d. What is this rate?

P'(t) at the time found in part (c)
...

3. Thanks for helping, but could you please show me how to arrive at the answers for both parts c and d.

Thank you

4. Originally Posted by Chris22
Thanks for helping, but could you please show me how to arrive at the answers for both parts c and d.
show what you get for P''(x)

5. For the first Derivative I got $\frac{224e^{5-t}}{(1+e^{5-t})^2}$

For the 2nd derivative I got $\frac{-224e^{5-t}+224e^{5-t}(e^{5-t})^4}{(1+e^{5-t})^2}$

Thats as far as I could get for the 2nd derivative. Any help would be greatly appreciated.

6. Factor the numerator by 224e^(5-t), difference of squares.
$\frac{-224e^{5-t}+224e^{5-t}(e^{5-t})^4}{(1+e^{5-t})^2}$
$\frac{224e^{5-t}{((e^{5-t})^4-1)}}{(1+e^{5-t})^2}$
$\frac{224e^{5-t}{((e^{5-t})^2-1)}{((e^{5-t})^2+1)}}{(1+e^{5-t})^2}$