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Ilya [14]
3 years ago
9

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.
Mathematics
1 answer:
alisha [4.7K]3 years ago
4 0

Answer:

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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Also,

\csc \theta=\frac{1}{\sin\theta}=\frac{1}{\sin (236.44)}=-\frac{6}{5}\text{ QED}

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