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22 February, 06:45

Collins body metabolizes caffeine at a rate of 14.3% per hour (so the amount of caffeine in Collins body decreases by 14.3% each hour). If Collin consumes a cup of coffee with 80 mg of caffeine in it, how long will it take for Collin's body to metabolize half of the 80 mg of caffeine?

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  1. 22 February, 07:54
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    It will take 4.5 h to metabolize 50% of the ingested caffeine.

    Explanation:

    Hi there!

    We have to find how long will it take Collin's body to metabolize 50% (40 g) of the ingested caffeine. We know that Collin's body metabolizes 14.3 % of the amount of caffeine each hour. Then, every hour the caffeine will be reduced by 14.3 %.

    After the first hour, the amount of caffeine will be:

    80 - 80 · 0.143 = 68.56

    Expressed in other form

    80 · (1 - 0.143) = 68.56

    80 · 0.857 = 68.56

    After the second hour, the amount of caffeine will be:

    68.56 - 68.56 · 0.143 = 58.75592

    or

    68.56 · 0.857 = 58.75592

    Since 68.56 = 80 · 0.857, we could write the amount of caffeine after 2 hours as:

    80. 0.857 · 0.857 = 58.75592

    After the third hour:

    58.75592 - 58.75592 · 0.143 = 50.35382344

    or

    58.75592 · 0.857 = 50.35382344

    In the same way, since 58.75592 = 80. 0.857 · 0.857 the amount of caffeine after 3 hours will be:

    80. 0.857 · 0.857 · 0.857 = 50.35382344

    Then after x hours, the amount of caffeine in the body will be:

    80 mg · 0.857ˣ

    We have to find the value of x for which that expression is equal to 40 mg:

    80 mg · 0.857ˣ = 40 mg

    0.857ˣ = 40 mg / 80 mg

    0.857ˣ = 0.5

    Apply ln to both side of the equation:

    ln (0.857ˣ) = ln (0.5)

    Apply logarithm property : ln (xᵃ) = a ln (x)

    x ln (0.857) = ln (0.5)

    x = ln (0.5) / ln (0.857)

    x = 4.5 h

    It will take 4.5 h to metabolize 50% of the ingested caffeine.
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