Question

The concentration, in μg / ml, of a drug introduced gradually into the body can be modeled by C(t)=5 t/0.04 t²+4.4. What is the rate of change in the concentration at t=40 ?

Answer 1

To find the rate of change of drug concentration at t=40, we need to differentiate the given function and then evaluate the derivative at t=40 using the quotient rule.

Step-by-step explanation:

The student is asking about the rate of change of a drug concentration in the body at a specific time, given by the function C(t)=5t/(0.04t²+4.4). To find the rate of change at t=40, we need to find the derivative of the function with respect to time (t) and evaluate it at t=40.

First, we differentiate C(t) using the quotient rule for derivatives:

The quotient rule is given by (v*u' - u*v') / v² for a function u/v, where u' and v' are the derivatives of u and v respectively.

Let u = 5t and v = 0.04t²+4.4. Then,
u'=5 and v' = 2*0.04*t = 0.08t. The derivative of C(t) is then given by (v*u' - u*v') / v²:

C'(t) = ((0.04t²+4.4)*5 - (5t)*0.08t) / (0.04t²+4.4)²

Now we evaluate C'(t) at t=40:

C'(40) = ((0.04*40²+4.4)*5 - (5*40)*0.08*40) / (0.04*40²+4.4)²

Now we simplify and calculate the result:

C'(40) = ((0.04*1600+4.4)*5 - 200*0.08*40) / (0.04*1600+4.4)²

C'(40) = ((64+4.4)*5 - 200*3.2) / (64+4.4)²
After simplifying further, we will obtain the rate of change for the drug concentration at t=40 seconds.