SOLUTION: Researcher are recording how much of an experimental medication is in a person’s bloodstream every hour. They discover that half-life of the medication is about 6 hours. Write

Question 1192444: Researcher are recording how much of an experimental medication is in a person’s bloodstream every hour. They discover that half-life of the medication is about 6 hours.
Write an equation to model how much medication will be in the bloodstream after an unknown number of days for an initial dose of 𝑎.?
Calculate how much medication is in a person’s bloodstream after 4 days if they took an initial dose of 500mg?
How much more medication will be in a person’s bloodstream if their initial dose was 750mg?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
if the initial dose is a, then the half life will be 1/2 * a.
the formula for half life becomes:
1/2 * a = a * g ^ 6
g is the growth rate.
divide both sides of the equation by a to get:
1/2 * a / a = g ^ 6
simplify to get:
1/2 = g ^ 6
solve for g to get:
g = (1/2) ^ (1/6) = .8908987181.
that's the growth rate per hour.
in 6 hours, a * .8908987181 ^ 6 = .5 * a.
in other words, the life of the mediation in your bloodstream is now half of what it was when the medicine was first administered.

in terms of days, the equation becomes:
y = a * g ^ (24 * d).
this tells you how much life is left after d days.

to see if this formula is correct, look for the half life of 6 hours in terms of days.
since 6 hours is .25 days, you get:
y = a * g ^ (24 * .25) which becomes:
y = a * g ^ 6.
since g is equal to .8908987181, this formula becomes:
y = a * .8908987181 ^ 6 = .5a.

the formula translated to days is:
y = a * g ^ (24 * d), where d is the number of days.

with an initial dose of 500 mg, the formula becomes:
y = 500 * g ^ (24 * 4) after 4 days.
simplify to get y = 500 * g ^ 96 = .0076293945.

with an initial dose of 750 mg, the formula becomes:
y = 750 * g ^ (24 * 4) after 4 days.
simplify to get y = 750 * g ^ 96 = .0114440918.

the amount of medication will be .0114440918 / .0076293945 = 1.5 times as much if they started with 750 mg rather than 500 mg.

you can figure this out from the equation.

with 500 mg start, the equation becomes 500 * g ^ (24 * 4)
with 750 mg start, the equation becomes 750 * g ^ (24 * 4)
(500 * g ^ (24 * 4)) / (750 * g ^ (24 * 4)) becomes 750 / 500 after g ^ (24 * 4) in the numerator and denominator cancel out.
750/500 = 1.5

the growth rate itself is g which is per hour.

you could have solved for the growth rate per days is you did the following.
you know the half life is in 6 hours.
6 hours is 6/24 = .25 days.
the formula for half life becomes 1/2 = g ^ .25
solve for g to get:
g = (1/2) ^ (1/.25) = (1/2) ^ 4 = .0625.
this g is the growth rate per day.

half life in number of days becomes 1/2 = g ^ .25 = .0625 ^ .25 = .5.
you now have the half life in terms of days rather than hours.
the advantage of doing it this way is that the rest of the problem is in days, rather than hours.

in 4 days, 500 mg becomes 500 * .0625 ^ 4 = .0076293945.
in 4 days, 750 mg becomes 750 * .0625 ^ 4 = .0114440918.

these are the same answers you got before when the growth rate was per hour.
it winds up being cleaner when you solve for the growth rate per day rather than per hour.

either way gets you the same answer.

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