Half life question

Question

“The rate of elimination of caffeine from the human body varies greatly from individual to individual. Suppose that Alex drinks a 16-ounce coffee drink that contains about 310 mg of caffeine. His body eliminates about 13% of the caffeine every hour.

Approximately how long will it take for Alex’s body to eliminate half of the caffeine? (As mentioned above, this amount of time is called the caffeine’s “half-life.”)

Answer 1

Answer 2

Use the formula

m(t) = m₀ e^-kt

where

m(t) = mass of caffeine after t hours

m₀ = initial mass of caffeine consumed, 310 mg

k = constant

The rate of elimination of caffeine is 13% per hour. In 1 hour, the remaining mass of caffeine is

(1 - 0.13)m₀ = 0.87m₀

From the formula, obtain

0.87m₀ = m₀ e^-k(1)

ln(0.87) = -k

k = -ln(0.87) = 0.1393

When m = (1/2)m₀ , the time required is given by

m₀/2 = m0 e^-0.1393t

0.5 = e^-0.1393t

-0.1393t = ln(0.5)

t = -ln(0.5)/0.1393 = 4.976 h

Answer: 4.98 hours (nearest 100^th). This is the half-life.