Exponential Decay

Question

A patient is given anesthesia for a medical procedure. An equation that can model the concentration of anesthesia in the bloodstream after t hours is:

f(t)=Ae^-kt

^{Assume that after 1 hour the concentration is 40mg and that after 3 hours the concentration is 25mg. Use this information to find the values of A and k for this example using algebraic methods}

Answer 1

You have two unknowns: The constants A and k. You therefore need 2 equations.

Since you are given 2 pairs of values for t and f(t), you can write these equations:

f(t)=Ae^-kt f(1) = 40, and f(3) = 25

40 = Ae^-k*1

25 = Ae^-k*3

Solving the pair of equations: It looks like taking the ln of each side might work:

ln(40) = ln(A) - k

ln(25) = ln(A) - 3k

Subtract:

ln(40) - ln(25) = 2k

Solve this for k, and then substitute in one of the equations to find A