c. Henry's body eliminated the antibiotic at half of the rate at which Greg's body eliminated the antibiotic.
To solve this, we are using the exponential decay function [tex]f(t)=a(1-b)^t[/tex]
where
[tex]f(t)[/tex] is the final amount remaining after [tex]t[/tex] hours
[tex]a[/tex] is the initial amount
[tex]b[/tex] is the decay rate in decimal form
[tex]t[/tex] is the time in hours
We know from our problem that Greg's body is getting rid of the medicine according to the function [tex]f(t)=200(0.88)^2[/tex]. We can find the decay rate by setting 0.88 equal to 1-b and solve for b:
[tex]1-b=0.88[/tex]
[tex]-b=0.88-1[/tex]
[tex]-b=-12[/tex]
[tex]b=0.12[/tex]
Since the rate is in decimal for, we are going to multiply it by 100% to express it as percentage:
Greg's body rate = 0.12*100% = 12%
Now, to find Henry's body rate, we are using the fact that when [tex]t=1[/tex], [tex]f(t)=282[/tex]. We can also infer that Henry's initial dose was 300 mg so [tex]a=300[/tex]. Let's replace the values in our decay function to find [tex]b[/tex]:
[tex]f(t)=a(1-b)^t[/tex]
[tex]282=300(1-b)^1[/tex]
[tex]\frac{282}{300} =1-b[/tex]
[tex]-b=\frac{282}{300} -1[/tex]
[tex]-b=-0.06[/tex]
[tex]b=0.06[/tex]
Henry's body rate = 0.06*100% = 6%
Since 6% is half of 12%, Henry's body eliminated the antibiotic at half of the rate at which Greg's body eliminated the antibiotic.
