A medicinal drug is administered into the bloodstream of a patient. Its concentration in the blood t minutes later, C(t), is measured in nanograms per milliliter (ng/mL). The drug company literature states that
C(t) = (400t)e^(-0.006t)
What is the peak concentration of the drug in the bloodstream (in ng/mL) and when does this occur?
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To find the peak concentration of the drug in the bloodstream, we need to find the maximum value of the function C(t) = (400t)e^(-0.006t).
Taking the derivative of C(t) with respect to t:
C'(t) = (400 - 2.4t)e^(-0.006t)
To find the critical points, we set C'(t) = 0:
(400 - 2.4t)e^(-0.006t) = 0
Setting each factor equal to zero:
400 - 2.4t = 0 => t = 400/2.4 = 166.67
e^(-0.006t) = 0 (impossible since e^x is always positive)
Therefore, the critical point occurs at t = 166.67.
To check if this point is a maximum or minimum, we can take the second derivative:
C''(t) = (2.4^2t - 400 * 2.4)e^(-0.006t)
Plugging in t = 166.67:
C''(166.67) = (2.4^2 * 166.67 - 400 * 2.4)e^(-0.006 * 166.67)
C''(166.67) ≈ -137.68
Since the second derivative is negative, this indicates a maximum.
Therefore, the peak concentration of the drug in the bloodstream is given by C(166.67).
C(166.67) ≈ (400 * 166.67)e^(-0.006 * 166.67) ≈ 52966.88 ng/mL
So, the peak concentration of the drug in the bloodstream is approximately 52,966.88 ng/mL and it occurs at approximately t = 166.67 minutes.
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