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3 November, 15:11

For a certain drug, the rate of reaction in appropriate units is given by Upper R prime (t) equalsStartFraction 2 Over t plus 1 EndFraction plus StartFraction 1 Over Start Root t plus 1 End Root End Fraction where t is time (in hours) after the drug is administered. Find the total reaction to the drug over the following time periods.

a. From t=1 to t=12.

b. From t=12 to t=24

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  1. 3 November, 16:33
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    a) 8.13

    b) 4.10

    Step-by-step explanation:

    Given the rate of reaction R' (t) = 2/t+1 + 1/√t+1

    In order to get the total reaction R (t) to the drugs at this times, we need to first integrate the given function to get R (t)

    On integrating R' (t)

    ∫ (2/t+1 + 1/√t+1) dt

    In integration, k∫f' (x) / f (x) dx = 1/k ln (fx) + C where k is any constant.

    ∫ (2/t+1 + 1/√t+1) dt

    = ∫ (2/t+1) dt + ∫ (1/√t+1) dt

    = 2∫ 1/t+1 dt + ∫1 / + (t+1) ^1/2 dt

    = 2ln (t+1) + 2 (t+1) ^1/2 + C

    = 2ln (t+1) + 2√ (t+1) + C

    a) For total reactions from t = 1 to t = 12

    When t = 1

    R (1) = 2ln2 + 2√2

    ≈ 4.21

    When t = 12

    R (12) = 2ln13 + 2√13

    ≈ 12.34

    R (12) - R (1) ≈ 12.34-4.21

    ≈ 8.13

    Total reactions to the drugs over the period from t = 1 to t = 12 is approx 8.13.

    b) For total reactions from t = 12 to t = 24

    When t = 12

    R (12) = 2ln13 + 2√13

    ≈ 12.34

    When t = 24

    R (24) = 2ln25 + 2√25

    ≈ 16.44

    R (12) - R (1) ≈ 16.44-12.34

    ≈ 4.10

    Total reactions to the drugs over the period from t = 12 to t = 24 is approx 4.10
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