Question

The concentration C (in mg/dl) of a certain antibiotic in a patients bloodstream is given by
C= 50t t2 + 25
where t is the time (in hours) after taking the antibiotic.
a. What is the concentration 4 hours after taking the antibiotic?
b. In order for the antibiotic to be effective, 4 or more mg/dl must be present in the bloodstream. When do you have to take the antibiotic again?

Answer 1

a)

C= 50t/( t^2 + 25)

To find  concentration 4 hours after taking the antibiotic, substitute t =4

C= 50*4/(4^2 + 25)

C= 200/(16+25)

C= 200/41

C=4.878  mg/dl

b)

4 or more mg/dl must be present, so

C>=4

50t/( t^2 + 25) >=4

50t >=4( t^2 + 25)

50t>=4t² +100

4t² -50t +100 <=0

2(2t² -25t+50) <=0

2(2t-5) (t-10)<=0

5/2<=t<=10

which means t should be less than or equals to 10.

you would need to take the antibiotic every 10 hours