Answer:
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.
