Final answer:
To find the maximum drug concentration and the time it occurs, we differentiate the concentration function K(x), set the derivative equal to zero to find critical points, and then evaluate K(x) at these points to find the maximum value. Option A is the correct answer.
Step-by-step explanation:
The student is asking about finding the maximum concentration of a drug in the bloodstream over time, which is a calculus problem involving the function K(x) = 5x / (x + 4). To determine the time at which the concentration is maximum, we need to find the derivative K'(x) and solve for when it equals zero, which will give us the critical points. To find the maximum concentration, we will evaluate the function K(x) at the critical points and determine the highest value.
To find the maximum concentration and the time at which it occurs, we first differentiate the function K(x) with respect to x to get K'(x), and then find the critical point by setting K'(x) = 0 and solving for x. After finding the critical value(s) for x, we test these values to see which one gives the maximum concentration by either evaluating the second derivative or using a sign chart. Finally, we plug the value of x that gives the maximum concentration into the original function K(x) to obtain the maximum concentration percentage.
In summary, the concentration will be maximized at x = (to be found after differentiation), and the maximum concentration is (to be calculated after finding x) % after x hours.