U3.L6.HW: Rational and Quadratic Inequalities: Question 13

Question

Answer the blank space, then explain your answer afterwards.

Following surgery, a patient has been receiving a pain relief medication intravenously. The concentration C (in milligrams per liter) of the medication in the patient's bloodstream t hours after this process started is given by 

C(t) = 30t + 6 / t + 3, t ≥ 0 (Note: The slash punctuation is in lieu of a fraction bar: C(t) = 30t + 6 is the numerator, while t + 3 is the denominator.)

The patient will not receive pain relief unless the concentration of the medication is 20 or more milligrams per liter. Use this function to determine the time interval for which the concentration of the medication, C , will be greater than or equal to 20 milligrams per liter.

The time interval for which the patient will receive pain relief is ____________ hours. (Express your answer in interval notation and round any decimals to the nearest tenth of an hour).

Answer 1

20 = (30t + 6)/(t + 3) we now want to solve with respect to t

20(t + 3) = 30t + 6

20t + 60 = 30t + 6

20t = 30t - 54

-10t = -54

t = 5.4 hours which is the minimum time upon which pain relief will set in

the time interval where the medication's concentration will be 20 or more milligrams per liter is thus [5.4, infinity) since this function always increases the more we move in the positive direction along t.