Question

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?

Answer 1

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.