Answer:
a) The antibiotic is made with a greater initial amount of active ingredient
b) The two medications will have the same amount of active ingredients in the blood stream at exactly 8 hours after first taking the medication.
Step-by-step explanation
Active ingredient of Nasal spray left in the bloodstream is modelled as
n(t) = ((t + 1)/(t + 5)) + (18/(t² + 8t + 15))
Active ingredient of Antibiotic left in the bloodstream is modelled as
a(t) = 9/(t+3)
with t in hours.
a) At the instance of usage, the active ingredient for both drugs are at their initial concentration as they appear in the drugs.
So, we find the amount of active ingredient for each drug in the bloodstream at t=0 and compare.
For nasal spray,
n(t) = ((t + 1)/(t + 5)) + (18/(t² + 8t + 15))
At t = 0
n(0) = (1/5) + (18/15) = 1.4
For antibiotics,
a(t) = 9/(t+3)
At t=0
a(0) = (9/3) = 3
3 > 1.4
Hence, the antibiotic is made with a greater initial amount of active ingredient.
b) when both medications will have the same amount of active ingredients in the bloodstream, n(t) = a(t)
n(t) = ((t + 1)/(t + 5)) + (18/(t² + 8t + 15))
a(t) = 9/(t+3)
Note that (t² + 8t + 15) = (t + 3)(t + 5)
When n(t) = a(t)
((t + 1)/(t + 5)) + (18/(t² + 8t + 15)) = 9/(t+3)
Multiplying through by (t² + 8t + 15) or more properly written as (t + 3)(t + 5)
(t+1)(t+3) + 18 = 9(t+5)
t² + 4t + 3 + 18 = 9t + 45
t² - 5t - 24 = 0
Solving the quadratic equation
t = 8 or t = -3
Time can't be negative, hence, t = 8 hours.
Hope this Helps!!!
