The concentration of a drug in a patient's bloodstream t hours after it is taken after it is taken is given by
C(t) = 0.016t/(t+2)^2 mg/cm^3
Find the maximum concentration of the drug and the time at which it occurs.
The maximum concentration of the drug occurs when the derivative of C(t) is equal to 0.
C'(t) = 0.016(2t+2)/(t+2)^3
0 = 0.016(2t+2)/(t+2)^3
2t+2 = 0
t = -1
The maximum concentration of the drug occurs at t = -1 and is 0.016 mg/cm^3.
To find the maximum concentration of the drug and the time at which it occurs, we need to determine the critical points of the concentration function and then analyze these points.
Step 1: Find the derivative of C(t)
The derivative of C(t) with respect to t will give us the rate of change of the concentration function.
C'(t) = [0.016(t+2)^2 - 2(0.016t)(2(t+2))]/(t+2)^4
= [0.016(t^2 + 4t + 4) - 0.064t(t+2)]/(t+2)^4
= (0.016t^2 + 0.064t + 0.064)/(t+2)^3
Step 2: Set the derivative equal to zero and solve for t
To find the critical points, we need to set the derivative equal to zero and solve for t.
(0.016t^2 + 0.064t + 0.064)/(t+2)^3 = 0
Multiply both sides by (t+2)^3 to get rid of the denominator:
0.016t^2 + 0.064t + 0.064 = 0
Step 3: Solve the quadratic equation
Solve the quadratic equation by factoring, completing the square, or using the quadratic formula.
Using the quadratic formula, we have:
t = (-0.064 ± √(0.064^2 - 4(0.016)(0.064)))/(2(0.016))
= (-0.064 ± √(0.004096 - 0.004096))/(0.032)
= (-0.064 ± √0)/(0.032)
= -0.064/0.032
= -2
So, t = -2 is the only critical point.
Step 4: Determine the maximum concentration
To determine if the critical point at t = -2 corresponds to a maximum concentration, we can analyze the behavior of the derivative on either side of the critical point.
When t < -2 (e.g., t = -3), the derivative will be negative, indicating a decreasing concentration.
When t > -2 (e.g., t = -1), the derivative will be positive, indicating an increasing concentration.
Therefore, at t = -2, we have a maximum concentration.
Step 5: Calculate the maximum concentration
To find the maximum concentration, substitute the critical point into the concentration function C(t).
C(-2) = 0.016(-2)/((-2)+2)^2
= 0
Therefore, the maximum concentration of the drug is 0 mg/cm^3 and it occurs at t = -2 hours.
To find the maximum concentration of the drug and the time at which it occurs, we need to find the derivative of the concentration function and solve for when the derivative equals zero.
Step 1: Differentiate the concentration function
To find the derivative of C(t), we can apply the quotient rule to differentiate 0.016t/(t+2)^2:
C'(t) = [0.016(t+2)^2 - 2(0.016t)(t+2)] / (t+2)^4
Simplifying this expression, we get:
C'(t) = [0.016(t^2 + 4t + 4) - 0.032t(t+2)] / (t+2)^4
C'(t) = [0.016t^2 + 0.064t + 0.064 - 0.032t^2 - 0.064t] / (t+2)^4
C'(t) = (0.016t^2 - 0.032t^2 + 0.064t - 0.064t + 0.064) / (t+2)^4
C'(t) = (0.016t^2 - 0.032t^2) / (t+2)^4
C'(t) = -0.016t^2 / (t+2)^4
Step 2: Find when the derivative equals zero
To find the time when the concentration is maximized, we need to find when the derivative equals zero. So, we solve the equation:
C'(t) = -0.016t^2 / (t+2)^4 = 0
Since the numerator will always be negative, the only way the fraction can equal zero is if the denominator equals zero:
t+2 = 0
Solving this equation, we find:
t = -2
Step 3: Determine the maximum concentration
To find the maximum concentration, we substitute the value of t = -2 into the concentration function:
C(-2) = 0.016(-2)/((-2)+2)^2 = 0 mg/cm^3
So, the maximum concentration of the drug is 0 mg/cm^3 and it occurs at t = -2.
Please note that it is unusual to have a negative time value in this context, so it's important to review the problem or double-check the calculations to ensure the accuracy of the results.