Answer to Question #121415 in Algebra for queen

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?

b. If the patient does not take another pill, how much of the pill will the patient use after an infinite amount of time? (Round answer to the nearest hundredth.)

Answer 1

Create a table listing the time elapsed (in hours), and the mass of the pill (in mg). List some points starting at t = 0.

At t = 0, the mass (m) of the pill is 50 mg - there is no initial metabolization.

At t = 1, 55% of the mass of the pill is gone, meaning we have a mass of 50 mg - (0.55)(50 mg) = (0.45)(50 mg) = 22.5 mg of pill remaining after the first hour.

After the second hour, 55% of the mass of the now 22.5 mg pill goes away, so we get 22.5 mg - (0.55)(22.5 mg) = (0.45)(22.5 mg) = 10.125 mg remaining after the second hour. If you continue writing down the pattern in your table, you should get the expression m(t) = 50(0.45)^t that tells you how much of the pill remains in the body after t hours.

So, for t = 4 hours, we get m(4) = 50(0.45)⁴ = about 2.05 mg remaining. (ANSWER TO PART A)

For c), the patient uses up 55% of the pill after each hour, so u(t) = 50(0.45)^t-1(0.55) = 27.5(0.45)^t-1. where t is the number of hours that elapsed (t ≥ 1). (If t = 0, u(0) = 0 since there would be no metabolization)

We get a geometric series with r = 0.45 and t*1* = 27.5. Since r is less than 1, this sum converges (i.e. has a finite value), and it can be calculated using S*infinity* = t*1/(1 - r). (t1* is the first term in the geometric series you're analyzing)

Plug in the numbers to get S*infinity* = (27.5 mg)/(1-0.45), which is equal to 50.00 mg of pill used. (ANSWER TO PART B) - this does make sense because if the patient does not take other pills, then all of the 50 mg pill would have fully dissolved into his bloodstream as time approaches infinity. (i.e. all 50 mg of the pill have been used