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 300 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) = 300(0.88)t

Which statement best describes the rate at which Henry's and Greg's bodies eliminated the medicine?
Henry's body eliminated the antibiotic faster than Greg's body.
Henry's body eliminated the antibiotic at the same rate as Greg's body.
Henry's body eliminated the antibiotic at half of the rate at which Greg's body eliminated the antibiotic.
Henry's body eliminated the antibiotic at one-fourth of the rate at which Greg's body eliminated the antibiotic.

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

To compare the rate at which Henry's and Greg's bodies eliminate the medicine, we need to look at the rate of change of the amount of medicine in their bodies over time.

For Henry, we can calculate the rate of change by finding the differences between consecutive values of f(t) and dividing by the corresponding time interval.

For example, for the first and second hours, the rate of change is:

(265.08 mg - 282 mg) / (2 hours - 1 hour) = -16.92 mg/hour

Similarly, we can calculate the rates of change for the rest of the time intervals:

(249.18 mg - 265.08 mg) / (3 hours - 2 hours) = -15.9 mg/hour
(234.22 mg - 249.18 mg) / (4 hours - 3 hours) = -14.96 mg/hour
(220.17 mg - 234.22 mg) / (5 hours - 4 hours) = -14.05 mg/hour

Now, let's consider Greg's formula f(t) = 300(0.88)^t. This formula gives the amount of medicine in Greg's body after time t.

To find the rate of change for Greg's body, we can take the derivative of the function f(t) with respect to t. The derivative of f(t) = 300(0.88)^t is:

f'(t) = 300 * ln(0.88) * (0.88)^t

Evaluating the derivative at t = 1, 2, 3, 4, and 5 hours, we get the following rates of change:

f'(1) = 300 * ln(0.88) * (0.88)^1 ≈ -21.94 mg/hour
f'(2) = 300 * ln(0.88) * (0.88)^2 ≈ -19.32 mg/hour
f'(3) = 300 * ln(0.88) * (0.88)^3 ≈ -17.02 mg/hour
f'(4) = 300 * ln(0.88) * (0.88)^4 ≈ -14.99 mg/hour
f'(5) = 300 * ln(0.88) * (0.88)^5 ≈ -13.18 mg/hour

Comparing the rates of change, we see that Henry's body eliminates the medicine at a faster rate than Greg's body. Specifically, for every hour, Henry eliminates the medicine at a faster rate, with a larger negative value, than Greg.

Therefore, the correct statement is: Henry's body eliminated the antibiotic faster than Greg's body.

To compare the rates at which Henry's and Greg's bodies eliminate the medicine, we need to compare the rate of change of the amount of medicine with respect to time for both of them.

For Henry's body, we can calculate the rate of change by finding the difference in the amount of medicine over a certain time interval. Let's choose the time interval from t = 1 to t = 2 hours:

Rate of change for Henry = (f(2) - f(1))/(2 - 1) = (265.08 - 282)/(2 - 1) = -16.92 mg/hour

For Greg's body, we can find the rate of change by taking the derivative of the equation f(t) = 300(0.88)t with respect to time:

f'(t) = 300(0.88)t * ln(0.88) = 264t * ln(0.88)

To compare the rates, let's calculate the rate of change for Greg's body at t = 1 hour:

Rate of change for Greg at t = 1 = f'(1) = 264 * 1 * ln(0.88) ≈ -20.74 mg/hour

Comparing the rates of change, we can see that the rate of elimination for Henry is -16.92 mg/hour, whereas the rate for Greg is -20.74 mg/hour.

Therefore, Henry's body eliminated the antibiotic faster than Greg's body.