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1 February, 00:29

The concentration of a drug t hours after being injected is given by

C (t) = (0.3t) / (t^2+5). Find the time when the concentration is at a maximum. Give your answer accurate to at least 2 decimal places.

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  1. 1 February, 03:42
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    t = 2.236 hours

    or 2 hours 14.16 minutes.

    Step-by-step explanation:

    C = (0.3t) / (t^2 + 5) where C is the concentration of the drug.

    Finding the derivative using the Quotient rule:

    C' = (t^2 + 5) (0.3) - (0.3t) * 2t / (t^2 + 5) ^2

    C' = 0.3t^2 + 1.5 - 0.6t^2 / (t^2 + 5) ^2

    C' = - 0.3t^2 + 1.5 / (t^2 + 5) ^2

    C is at a maximum when the derivative is zero.

    Equating this to zero:

    -0.3t^2 + 1.5 / (t^2 + 5) ^2 = 0

    -0.3t^2 + 1.5 = 0

    -0.3t^2 = - 1.5

    t^2 = - 1.5 / - 0.3 = 5

    t = √5

    t = 2.236 hours.
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