Question

A patient takes a pill that holds 50 mg of medicine in it. The pill is metabolized, or used up, at a rate of 55% every hour.

a. Rounded to the nearest hundredth, how much of the pill has been used by the body after four hours?

Answer 1

After every hour, 55% of the pill is used up.

Initial mass = 50 mg.

After one hour, amount of pill used up:

= 55% of 50 mg

=0.55 $\times$ 50 mg

so, the amount of pill remaining after one hour = initial mass - used up mass

= 50 - 0.55 × 50 mg

= 50 (1-0.55) mg

= 50 x 0.45 mg

between one and two hours, amount of pill used up:

= 55% of 50 x 0.45 mg

=0.55 × 50 x 0.45 mg

so, the amount of pill remaining after two hours = initial mass - used up mass

= 50 x 0.45 - 0.55 × 0.45 x 50 mg

= 50 x 0.45(1-0.55) mg

= 50 x 0.45² mg

similarly amount of pill remaining at the end of nth hour = 50 x 0.45ⁿ mg

so, the amount of pill remaining at the end of 4th hour = 50 x 0.45⁴ mg

the amount of pill used up after 4 hours = 50 - 50 x 0.45⁴ mg

= 47.95 mg