Mathematics: The concentration of a drug in the bloodstream C(t) at any time t, in hours, is described by the equationC(t)=100t/t^2+25where t=0 corresponds to the time at which the drug was swallowed. Determine how long it takes the drug to reach its maximum concentration.It will take hours until it reaches its maximum concentration.

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The concentration of a drug in the bloodstream C(t)

coco9972

23.01.2022

It will take 5 hours until it reaches its maximum concentration.

Step-by-step explanation:

The maximum concentration will happen in t hours. t is found when

C'(t) = 0

In this problem

$C(t) = \frac{100t}{t^{2} + 25}$

Applying the quotient derivative formula

$C'(t) = \frac{(100t)'(t^{2} + 25) - (t^{2} + 25)'(100t)}{(t^{2} + 25)^{2}}$

$C'(t) = \frac{100t^{2} + 2500 - 200t^{2}}{(t^{2} + 25)^{2}}$

$C'(t) = \frac{-100t^{2} + 2500}{(t^{2} + 25)^{2}}$

A fraction is equal to zero when the numerator is 0. So

$-100t^{2} + 2500 = 0$

$100t^{2} = 2500$

$t^{2} = 25$

$t = \pm \sqrt{25}$

$t = \pm 5$

We use only positive value.

It will take 5 hours until it reaches its maximum concentration.

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