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Math Help - Help with Derivatives in Word Problems

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by Chris22 View Post
    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)
    ...
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  3. #3
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    Thanks for helping, but could you please show me how to arrive at the answers for both parts c and d.

    Thank you
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  4. #4
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    Quote Originally Posted by Chris22 View Post
    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)
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  5. #5
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    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.
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  6. #6
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    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}
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