Help with Derivatives in Word Problems

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October 27th 2009, 05:06 PM
Chris22

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?
October 27th 2009, 05:25 PM
skeeter

Quote:

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)

...
October 27th 2009, 05:32 PM
Chris22

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

Thank you
October 27th 2009, 05:55 PM
skeeter

Quote:

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)
October 27th 2009, 06:34 PM
Chris22

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.
October 27th 2009, 06:41 PM
Lord Voldemort

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