Anonymous
Anonymous asked in Science & MathematicsMathematics · 4 weeks ago

PLEASE HELP!!!!!!!?

Many soft drinks contain about 40 mg of caffeine in one can. Every 5 hours, the mass of caffeine in an

adult’s bloodstream reduces by half.

If there currently is 2.5mg of caffeine in a person’s bloodstream, how many hours ago did she consume the

soft drink? An algebraic solution is required for full marks.

3 Answers

Relevance
  • 4 weeks ago
    Favourite answer

    t = 0 => 40mg

    t = 5 => 20mg....time to halve

    t = 10 => 10mg

    t = 15 => 5mg

    t = 20 => 2.5mg

    i.e. if 2.5mg is in the bloodstream, the caffeine was consumed about 20 hours previously.

    We could model this with the function C(t) = 40(0.5)ᵗ/⁵

    Hence, we require when 2.5 = 40(0.5)ᵗ/⁵

    i.e. (0.5)ᵗ/⁵ = 1/16

    or, (1/2)ᵗ/⁵ = (1/2)⁴

    Hence, t/5 = 4

    so, t = 20 hours as before

    :)> 

    • Commenter avatarLog in to reply to the answers
  • Ash
    Lv 7
    4 weeks ago

    Using half life formula

    N = N₀(½)^(t / t½).........where t½ = half life

    Given t½ = 5 hours

    N = 2.5 mg

    N₀ = 40 mg

    2.5 = 40 (½)^(t/5)

    (½)^(t/5) = 2.5/40

    (½)^(t/5) = 1/16

    ln (½)^(t/5) = ln(1/16)

    (t/5) ln (½) = ln(1/16)

    t = 5 ln(1/16) / ln (½)

    t = 20 hours

    • Commenter avatarLog in to reply to the answers
  • 4 weeks ago

     Given the half-life, the rate of decay may be found.

    0.5 = e^(5k)

    ln (0.5) = 5k

    k= -0.139

     

    40-2.5 =37.5 

     

    Using the rate, we can find the time it takes for the amount to decay 37.5 mg of its mass.

    2.5/40 = e^(-0.139t)

    ln(2.5/40) = -0.139t

    ln(.0625) = -0.139t

    t=19.95 hours 

    • Commenter avatarLog in to reply to the answers
Still have questions? Get answers by asking now.