Question

the amount of medication, in milligrams, in a patient's bloodstream after t hours, can be represented by the following function: m of t equals 88 over the quantity 1 plus e to the x plus 7 tenths power end quantity what is the rate of change for the amount of medication in the patient's bloodstream after 4 hours?

Answer 1

about -0.79 mg/hour

Explanation:

You want the rate of change of the amount of medication in a patient after 4 hours if the amount after t hours is modeled by y=88/(1+e^(t+0.7)).

Rate of change

The rate of change is found by taking the derivative of the function with respect to time. Let u = 1+e^(t+0.7). Then du/dt = e^(t +0.7).

The function can be written ...

y = 88/u = 88·u^-1

so its derivative is ...

dy/dt = (-88·u^-2)·du/dt

dy/dt = -88(e^(t +0.7))/(1 +e^(t +0.7))^2

At t=4

The value of the rate of change at t=4 is ...

dy/dt = -88(e^(4 +0.7))/(1 +e^(4 +0.7))^2 ≈ -88(109.95)/(1 +109.95)^2

dy/dt ≈ -0.78602 ≈ -0.79 . . . . . . mg/h

The rate of change in the amount of medication is about -0.79 mg/h.