Question
Asked Oct 24, 2019
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The concentration, y, of a drug in the bloodstream t seconds after injection into a muscle is given by

y = 3(e−7t − e−9t) for t ≥ 0.

The concentration is maximum after t= ? seconds.

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Expert Answer

Step 1

Given,

           The concentration, y, of a drug in the bloodstream t seconds after injection into a muscle is given by

                          y = 3(e−7t − e−9t) for t ≥ 0.

Now, differentiating on both sides, we get

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a(3(e-7t -e-3)) dy y' dx dx - 3 d(e-7t - e-9t) dx = 3 d(e-7t) dx 3 de-9t) dx 3 Зе-7t(-7) — Зе -9t(-9) =-21e-7t 27e-9t Now finding the critical point by equating y' 0. -9t 0 .-21e-7t 27e -21e-7t -27e-9t 7e-7t = 9e-9t

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Step 2

Taking ln on both sides, we get

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In(7e7 n(9e-9t) In(7) ne7t) In(9) n(e9t) In(7) 7t In(e) = In(9)- 9t In(e) In(7) 7t In(9) 9t 9t 7t ln(9) - In (7) 2t n(9)In (7) tn t0.125657

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Step 3

Now, finding the 2nd deriv...

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d-21e-7t+27e-9t) dy у" dx dx = -21e-7(-7) 27e-9 (-9) 147e-7t 243e-9t = Since, y"0.125657 147e7(0.125657)_ 243e-9(0.125657) -17.249 < 0 So, the concentration is maximum after t = ln or 0.125657

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Tagged in

Math

Calculus

Derivative