The amount (in milligrams) of a drug in a person's body following one dose is given by an exponential decay function. Let () denote the amount of drug in the body at
time in hours after the dose was taken. In addition, suppose you know that (3) =
22.7 and (6) = 15.2.
a. Find a formula for in the form () =

, where you determine the values
of and exactly.
b. What is the size of the initial dose the person was given?
c. How much of the drug remains in the person's body 8 hours after the dose
was taken?
d. Estimate how long it will take until there is less than 1 mg of the drug
remaining in the body.
e. Compute the average rate of change of on the intervals [3,5],
[5,7], and [7,9]. Explain the meaning of the values you found, including
appropriate units. Write at least one additional sentence to explain any
overall trend(s) you observe in the average rate of change.

1. when "a" is a root of a polynomial, (x -a) is a factor of it. for the three roots given, the factors of the desired polynomial are (x +2)(x -1/2)(x -3).

in order to make the leading coefficient be negative, we need to multiply this product by a negative number. any negative number will do, but we choose a small (magnitude) value that will eliminate the fraction: -2.

then

f(x) = -2(x +2)(x -1/2)(x -3) = -(x +2)(2x -1)(x -3)

= -(2x² +3x -2)(x -3)

= -(2x³ -3x² -11x +6)

f(x) = -2x³ +3x² +11x -6

2. a graph created by the desmos on-line graphing calculator is shown, and the zeros are highlighted.

3. as indicated in part 1, the multiplier of this equation can be anything and the zeros will remain the same. you want a negative leading coefficient, so the "anything" is restricted to any of the infinite number of numbers that will make that be the case.

step-by-step explanation: