Question

estigating Patterns in Data 2 Some oral medications help us get better by entering our bloodstream. After we take the amount of the medicine that is in our bloodstream will decrease over time We can model this decrease by an exponential curve. The following table shows the amount of is a common brand name) in a person's bloodstream t hours after a person takes the table and graph below to answer the following questions acetaminophen (Tylenol he medicine. Use Amount of 700 (in hours) Acetamnophen600 (in mg) S 500 650 400 6 300 305 200 100 65 28 Time (in hours) 12 650 (0.46)* predicts the amount of acetaminophen in a person's A bloodstream x hours after taking the drug. What is the initial amount of acetaminophen in the person's bloodstream? The initial amount in the person's bloodstream is the amount of drug that the person took. The exponential model B When will the amount of acetaminophen be below 100 mg? 650(0.45)* predicts the amount of acetaminophen in a person's The exponential curve c bloodstream x hours after taking the drug. Use this equation to predict the amount of acetaminophen in a person's bloodstream 3.5 hours after he first took the medicine. What is the value of b in the exponential curve? Does this indicate exponential growth or exponential decay? D

Answer 1

A.

The exponential model is,

$\hat{y}$ = 650 (0.46)^x

Initial amount in the blood is at x = 0

= 650 (0.46)⁰

= 650 mg

B.

When amount of acetaminophen is below 100 mg

$\hat{y}$ < 100

650 (0.46)^x < 100

(0.46)^x < (100/650)

(0.46)^x <  0.1538

Taking log,

x log(0.46) < log(0.1538)

-0.7765 x < -1.872

0.7765 x > 1.872

=> x > 1.872/ 0.7765

=> x > 2.41

Thus, by the exponential model, the amount of acetaminophen is below 100 mg after 2.41 hours

(c)

At x = 3.5

Predicted acetaminophen, $\hat{y}$ = 650 (0.46)^3.5

= 42.91 mg

(d)

The value of b in the exponential curve is 0.46

Since the value of b is less than 1, the model is an exponential decay (as the acetaminophen in the bloodstream decays with time x)