Could someone please help me with this problem?!?

Question

The concentration C(t) in milligrams per cubic centimeter of a medication in a patients blood is given by C(t) = 0.2t/(2t + 3)^2, where t is the number of hours have the medication is taken. How long does it take for the maximum concentration to be present in the patient?

a) 0.50 h
b) 1.50 h
c) 0.25 h
d) 0.67 h

Answer 1

You need to take the first derivative of C(t), set it equal to zero, and solve. This will give the values of t for which C(t) has an extreme value. You must then inspect them to determine which is the maximum.

C(t) = 0.2t / (2t + 3)^2 = 0.2t / (4t^2 + 12t + 9)
C'(t) = ((4t^2 + 12t + 9)*0.2 - 0.2t(8t + 12)) / (2t + 3)^4
= (-0.8t^2 + 1.8) / (2t + 3)^4

To solve for this expression equal to zero, just solve for the numerator equal to zero. That's -0.8t^2 + 1.8 = 0 ==> t^2 = 2.25 ==> t = +/- 1.5. However, using t = -0.15 makes no sense since we're talking about a forwards progression of time, not to mention it returns an undefined value for C(t) and C'(t). So it must be t = 1.5 h, answer B).

Answer 2

The previous answers are correct and were solved by calculus. I graphed the equation using Excel and got a max concentration of 0.00833 at 1.5 hours.