Course:MATH102/Question Challenge/2008 December Q2b

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Question

The following problems require little or no computation. Put the answer in the box provided. You may use space here for supporting reasoning or computations, if any, for which part-marks would be awarded.

(b) A differential equation describing radioactive decay of a substance is

$\displaystyle {\frac {dy}{dt}}=-ky,$

where $y(t)=Ce^{-kt}$ is the amount of radioactivity remaining. A plot of $y(t)$ versus t is found to be a curve that goes through the point $(0,10)$ and has tangent line of slope −5 at that point. What are the values of the constants?

$C=$

$k=$

Hints

Hint
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Solutions

Solution
The initial value is given by the point (0, 10) therefore: $C=10$ We are also told that at the point (0, 10) the tangent line has a slope of $-5$: ${\frac {dy}{dx}}=-ky$ $-5=-k(10)$ $k=5/10$ $k=1/2$