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January 29th, 2018, 09:26 AM   #1
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Question Applied integrals find the time

When Bob runs a 10km marathon he generates the power (P).

$\displaystyle \displaystyle P(t) = 600e^{-\frac{t}{5000}}$

Where the time is in seconds. A sugerpience contains about 40kJ energy. How long time does it take for Bob to run a 10 km if he consumes 30 sugarcubes during the marathon.

[IMG]Link to problem http://www.mediafire.com/view/u68i8n8zvmcn3hk/IMG_20180129_181555.jpg[/IMG]

What do I need to do to solve for t (the time)? I tried doing this first but that didn't work out well.

$\displaystyle
\displaystyle P(t) = 600e^{-\frac{t}{5000}}\\
\displaystyle E_{total} = E_s \cdot t\\
\displaystyle \int P(t) \ dt = E_s \cdot t\\
\displaystyle \int 600e^{-\frac{t}{5000}} \ dt = 1200\\
$

No solutions for real numbers.
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January 29th, 2018, 10:39 AM   #2
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$\displaystyle 600\int_0^t e^{-\frac{\tau}{5000}}d\tau= \left[-3000000 e^{-\frac{\tau}{5000}}\right]_0^t= -300000\left(1- e^{\frac{t}{5000}}\right)= 1200$.

$\displaystyle 1- e{\frac{t}{5000}}= -\frac{1200}{300000}= -\frac{12}{3000}= -0.004$

$\displaystyle e^{-\frac{t}{5000}}- 1= 0.004$

$\displaystyle e^{-\frac{t}{5000}}= 1.004$

$\displaystyle -\frac{t}{5000}= ln(1.004)= 0.00399$

$\displaystyle t= -20$

Is it that this is negative that bothers you?

Last edited by Country Boy; January 29th, 2018 at 10:52 AM.
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January 29th, 2018, 11:19 AM   #3
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I get t = -1.99600107. That doesn't seem realistic that Bob ran the 10 km marathon in about 2 seconds. I think you missed a 0 in -1200 / 30 0000 should be -1200 / 3 000 000 if I'm not mistaken.

Last edited by DecoratorFawn82; January 29th, 2018 at 11:22 AM.
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January 29th, 2018, 11:33 AM   #4
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$P = \dfrac{dW}{dt} = 600e^{-t/5000}$

$\displaystyle W = 600 \int_0^T e^{-t/5000} \, dt$

$30 \cdot 40\, kJ = 1200 \, kJ$

$\displaystyle 1200000 = 600 \int_0^T e^{-t/5000} \, dt$

$\displaystyle 2000 = \int_0^T e^{-t/5000} \, dt$

$2000 = 5000\left[1-e^{-T/5000}\right]$

$e^{-T/5000} = \dfrac{3}{5}$

$T = 5000 \log(5/3) \approx 2554 \, sec \approx 43 \, min$

reasonable ... about a 7 minute per mile pace
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