When a certain drug is taken orally, the concentration of the drug in the patients bloodstream after t minutes is given by C(t)=0.06t-0.0002t^2, where 0 is less than or equal to "t" which is less than or equal to 240 and the concentration is measured by mg/L. When is the maximum serum concentration reached, and what is that maximum concentration?
To find the maximum serum concentration and the time it occurs, we need to determine the peak of the concentration function C(t) = 0.06t - 0.0002t^2.
First, let's rewrite the equation in the standard quadratic form ax^2 + bx + c. In our case, a = -0.0002, b = 0.06, and c = 0.
Since we want to find the maximum, we know that the maximum value occurs at the vertex of the parabola represented by the equation. The x-coordinate of the vertex can be found using the formula x = -b / (2a).
In our case, x = -0.06 / (2 * -0.0002) = 150.
Now we substitute this value back into the original equation to find the maximum concentration, C(t):
C(t) = 0.06t - 0.0002t^2
C(150) = 0.06 * 150 - 0.0002 * 150^2
C(150) = 9 - 4.5
C(150) = 4.5 mg/L
Therefore, the maximum serum concentration is reached at t = 150 minutes, and the value of the maximum concentration is 4.5 mg/L.