Anonymous
Anonymous asked in Science & MathematicsMathematics · 4 weeks ago

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

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.

Relevance

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

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• 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

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•  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

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