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?

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.