Question

Sarah is fighting a sinus infection. Her doctor prescribed a nasal spray and an antibiotic to fight the infection. The active ingredients, in milligrams, remaining in the bloodstream from the nasal spray, n(t), and the antibiotic, a(t), are modeled in the functions below, where t is the time in hours since the medications were taken. n(t) = ((t + 1)/(t + 5)) + (18/(t² + 8t + 15)) a(t) = 9/(t+3)Determine which drug is made with a greater initial amount of active ingredient. Justify your answer. Sarah's doctor told her to take both drugs at the same time. Determine algebraically the number of hours after taking the medications when both medications will have the same amount of active ingredient remaining in her bloodstream.

Answer 1

a) The antibiotic is made with a greater initial amount of active ingredient.

b) The time when both medications will have the same amount of active ingredients in the bloodstream = 8 hours.

Explanation:

Nasal spray's active ingredient left in the bloodstream is modelled as

n(t) = ((t + 1)/(t + 5)) + (18/(t² + 8t + 15))

Antibiotic's active ingredient left in the bloodstream is modelled as

a(t) = 9/(t+3)

with t in hours.

a) Determine which drug is made with a greater initial amount of active ingredient. Justify your answer.

At the instance of usage, the active ingredient for both drugs are at their pure concentration.

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 evidently made with a greater initial amount of active ingredient.

b) Determine algebraically the number of hours after taking the medications when both medications will have the same amount of active ingredient remaining in her bloodstream

At the time 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 cannot be negative, hence, t = 8 hours.

Hope this Helps!!!