The number of malaria parasites (p) in the blood h hours after a patient takes the medication is modelled by the function: p = 200 / [0.9 + 0.1e^(2h)]

Question

The number of malaria parasites (p) in the blood h hours after a patient takes the medication is modelled by the function: p = 200 / [0.9 + 0.1e^(2h)]

a. what is the initial number of parasites in the blood?
b. how long did it take before the number of parasites had been halved?
c. after how many hours was the number of parasites decreasing the fastest?
d. what does the sketch of the function look like?
e. give a possible explanation for the shape of the curve

Answer 1

a) The initial number of parasites is when h = 0, so p = 200/[0.9+0.1] = 200 parasites.

b) Set p = 100, which is half of the parasites since it started.  Solve for h:

     100 = 200 / [0.9 + 0.1e^(2h)]

     1/2 = 1 / [0.9 + 0.1e^(2h)]

     2 = 0.9 + 0.1e^(2h)

     1.1 = 0.1e^(2h)

     11 = e^(2h)

     ln 11 = 2h

     1.199 hours = h

c) Take the derivative of p using MATH 8 and graph it, or take the derivative by hand using the quotient rule and graph it.  Find the minimum of the graph, which is where the rate of change is smallest (decreasing fastest) which is when h = 1.099 hours

d) I'm assuming you can use a graphing calculator to answer this question because getting values by hand is not feasible.  The graph looks like a decreasing exponential function with an asymptote at y = 0.

e) The number of parasites drops off sharply after the medication is taken, eventually slowing down and tapering off.