Hello, How would one solve this problem? thnk you

Question

The quantity of a drug in the blood stream t minutes after the tablet is swallowed is given, in mg, by

𝑄(𝑡) = 50𝑡𝑒^−.01𝑡

a.) At what time is the quantity of the drug in the bloodstream a maximum?

b.) What is that maximum?

Answer 1

Ayana,

You can use properties of the first derivative to find when the function is a maximum. Just find the critical points where Q'(t) is equal to 0 or undefined, and then determine if the point is a maximum by checking to see if Q'(t) changes from positive to negative at the point.

For part b, just plug in the time you found in part a into the equation for Q.

Answer 2

a) Let differentiate function Q(t): Q'(t) = 50e^{- 0.01t} - 0.5te^{- 0.01t}. To get critical number we set Q'(t) = 50e^{- 0.01t} - 0.5te^{- 0.01t}.= 0, so 50 - 0.5t = 0, from here t = 100 min. Let find Q''(t) = - e^{- 0.01t} _{+ 0.005te}^{- 0.01t};

So, Q''(100) = (0.005·100 - 1)e^{- 1} = -0.5/e < 0. By second derivative test we have max at t = 100 min

b) Q_max = Q(100) = 50·100e^{- 0.01·100·}= 5000/e = 1839.4 mg