Question

Henry is taking a medicine for a common cold. The table below shows the amount of medicine f(t), in mg, that was present in Henry's body after time t: t (hours) 1 2 3 4 5 f(t) (mg) 282 265.08 249.18 234.22 220.17 Greg was administered 200 mg of the same medicine. The amount of medicine in his body f(t) after time t is shown by the equation below: f(t) = 200(0.88)t Which statement best describes the rate at which Henry's and Greg's bodies eliminated the medicin

Answer 1

After one hour, Henry's body loses 282 - 265.08 = 16.92 mg of medicine, while Greg's body loses 200 - 200(0.88) = 24 mg of medicine

Therefore, Greg's body is removing the medicine faster.

Answer 2

c. Henry's body eliminated the antibiotic at half of the rate at which Greg's body eliminated the antibiotic.

Step-by-step explanation

To solve this, we are using the exponential decay function
`f(t)=a(1-b)^t`

where

`f(t)` is the final amount remaining after
`t` hours

`a` is the initial amount

`b` is the decay rate in decimal form

`t` is the time in hours

We know from our problem that Greg's body is getting rid of the medicine according to the function
`f(t)=200(0.88)^2`. We can find the decay rate by setting 0.88 equal to 1-b and solve for b:

`1-b=0.88`

`-b=0.88-1`

`-b=-12`

`b=0.12`

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
`t=1`,
`f(t)=282`. We can also infer that Henry's initial dose was 300 mg so
`a=300`. Let's replace the values in our decay function to find
`b`:

`f(t)=a(1-b)^t`

`282=300(1-b)^1`

`(282)/(300) =1-b`

`-b=(282)/(300) -1`

`-b=-0.06`

`b=0.06`

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.