# Chapter 3 Examples and Additional Exercises Chapter 3 Examples and Additional Exercises

Sections 3.1 and 3.2

1.The prescriber orders 15 milligrams per kilogram of a drug for a patient. The amount per milligram is given by the equation, where k is the patient's weight in kilograms. Complete the table.

Weight (kg) / Dosage (mg)
50
55
60
65
70
75
80
85
90
95
100

Then, plot the points on the coordinate system.

2. A mathematical model for the number of kidney transplants performed in the United States in the years 2001 through 2005 is given by, where y is the number of transplants and t represents the year, with t=1 corresponding to 2001.

(a) Complete the table.

Year / 2001 / 2002 / 2003 / 2004 / 2005
Transplants

(b) Determine whether each ordered pair is a solution to the equation:

(i). (0, 13,852)

(ii). (6,16,845)

(iii). (8, 18,859.68)

(iv). (2010, 20723)

(c) Plot the points from the table on a coordinate system. Connect the points with a smooth curve.

(d) Using this model, what is the prediction for the number of transplants in the year 2011? Do you think this prediction is valid? What factors could affect the model’s accuracy?

3. Suppose a patient's temperature over a period of time is graphed on a coordinate system, with temperature (T) on the y-axis and time (t) in hours on the x-axis.

(a) What do the x-intercepts of the graph represent? Would you expect to have any x-intercepts?

(b) What do the y-intercepts of the graph represent? Would you expect to have any y-intercepts?

4.

Fetal Station

This is the relationship between the presenting part, whether that be the head, shoulder, buttocks, or feet, and two parts of the maternal pelvis called the ischial spines. Normally the ischial spines are the narrowest part of the pelvis, and are a natural measuring point for the delivery progress. If the presenting part lies above the ischial spines, the station is reported as a negative number from 0 to -3 where each number is a centimeter. If the presenting part lies below the ischial spines, the station is reported as a positive number from 0 to 3 where each number is a centimeter. The baby is said to be "engaged" in the pelvis when it reaches 0 station.

Bishop's Score for Induction

This is the table used to determine how successful an induction of labor might be. It is recommended that the Bishop's Score by greater than 5 for induction to be successful.

Each component is given a score of 0-2 or 0-3. The highest possible score is 13.

Bishop score
Parameter\Score / 0 / 1 / 2 / 3
Position / Posterior / Mid-position / Anterior / -
Consistency / Firm / Intermediate / Soft / -
Effacement / 0-30% / 40-50% / 60-70% / 80%+
Dilation / <1 cm / 1-2 cm / 2-3 cm / >3 cm
Fetal station / -3 / -2 / -1, 0 / +1, +2

Interpretation: A score of 5 or less suggests that labor is unlikely to start without induction. A score of 9 or more indicates that labor will most likely commence spontaneously.

A low Bishop's score often indicates that induction is unlikely to be successful. Some sources indicate that only a score of 8 or greater is reliably predictive of a successful induction.

Modifiers

1. Add 1 point to score for:
2. Preeclampsia
3. Each prior vaginal delivery
4. Subtract 1 point from score for:
5. Postdates pregnancy
6. Nulliparity
7. Premature or prolonged rupture of membranes

Cesarean rates: first time mothers women with past vaginal deliveries

Scores of 0 – 3: 45% 7.7%
Scores of 4 - 6: 10% 3.9%
Scores of 7 - 10: 1.4% .9%

1. An examination shows a patient position is mid-position, consistency is soft, effacement is 60-70%, and dilation is 1-2 cm.

(a) Complete the table.

Fetal Station / -3 / -2 / -1 / 0 / 1 / 2
Bishop Score

(b) Plot the points from the table on the coordinate plane, with fetal station on the x-axis and bishop score on the y-axis.

(c) In order for induction to be reliably successful for this patient, what should the fetal position be?

Section 3.3

1. A person given a stress test is generally told that should the heart rate reach a certain point, the test will be stopped. The maximum allowable heart rate depends on the person’s age. The maximum allowable heart rate, m, in beats per minute, can be approximated by the equation, where x represents the patient’s age from 1 through 99. Using this mathematical model, find

(a) the maximum hear rate for a 50-year-old.

(b) the age of a person whose maximum heart rate is 160 beats per minute.

(c) the slope and y-intercept of the graph of the equation

(d) Interpret the slope and y-intercept in the context of the problem.

2. The concentration C (in milligrams per milliliter) of a drug in a patient’s bloodstream is monitored over 10-minute intervals for 2 hours, where t is measured in minutes, as shown in the table. Find the average rate of change over each interval.

a. [0,10]

b. [0,20]

c. [100,110]

t / 0 / 10 / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90 / 100 / 110 / 120

### C

/ 0 / 2 / 17 / 37 / 55 / 73 / 89 / 103 / 111 / 113 / 113 / 103 / 68

3.Weights and Drug Doses

The dosage chart below was prepared by a drug company for doctors who prescribed Tobramycin, a drug that combats serious bacterial infections such as those in the central nervous system, for life-threatening situations.

Weight (pounds) Usual Dosage (mg) Maximum Dosage (mg)

88 40 66

99 45 75

110 50 83

121 55 91

132 60 100

143 65 108

154 70 116

165 75 125

176 80 133

187 85 141

198 90 150

209 95 158

1. Use grid paper to plot the data (weight, usual dosage) and draw a best-fit line.

2. Plot (weight, maximum dosage) on the same axes. Draw a best-fit line.

3. Find the slope for each line. What do they mean, and how do they compare?

4. Write the equations of the two lines.

5. Are the two lines parallel? Why or why not?

Section 3.4

1. As seen in the following graph, the expected number of remaining years of life of a person, y, approximates a linear function. The expected number of remaining years is a function of the person’s current age, a, for. For example, from the graph we see that a person who is currently 50 years old has a life expectancy of 36.0 more years.

(a) Using the two points on the graph, determine the function y(a) that can be used to approximate the graph.

(b) Using the function from part (a), estimate the life expectancy of a person who is

currently 37 years old.

(c) Using the function from part (a), estimate the current age of a person who has a life expectancy of 25 years.

2. Deaths due to heart disease have been declining approximately linearly since the year 2000. The bar graph below shows the number of deaths, per 100,000 deaths, due to heart disease in selected years, projected for 2006-2010. See Graph.

(a) Let r be the number of deaths due to heart disease per 100,000 deaths, and let t represent the years since 2000. Write the linear equation that can be used to approximate the data.

(b) Use the equation from part (a) to estimate the death rate due to heart disease in 2006.

(c) Assuming this trend continues until the year 2020, estimate the death rate due to heart disease in 2020.

3. Each ordered pair gives the exposure index x of a carcinogenic substance and the cancer mortality y per 100,000 people in the population.

(3.50, 150.1), (3.58, 133.1), (4.42, 132.9), (2.26, 116.7), (2.63, 140.7), (4.85, 165.5), (12.65, 210.7), (7.42, 181.0), (9.35, 213.4)

(a) Plot the data. From the graph, do the data appear to be approximately linear?

(b) Visually find a linear model for the data. Graph the model.

(c) Use the model to approximate y if x = 3.

Section 3.5

1. The percentage of Americans 18 and over who smoke has been decreasing approximately linearly since 1997. In 1997, approximately 29 .2% of Americans 18 and older smoked. In 2004, approximately 20.9% of those 18 and older smoked.

(a) Draw a graph that fits these data.

(b) On the graph, darken the part of the graph where the percentage of Americans 18 and older is less than or equal to 25%.

(c) Estimate the first year that the percentage of Americans 18 and older was less than 23%.

2. The prescriber orders of a drug PO state.

(a) Find an equation for the total amount (mg) of the drug prescribed for a person with a given body surface area ().

(b) The "safe dose range" for this drug is 20 to 40 mg. per day. Graph the equation from part (a) and shade the portion of the graph that represents the "safe dose range".

(c) What range does the patient's BSA have to fall in for the dose to be in the safe range?

(d) Gary is 5'11" tall and weighs 175 pounds. Is he in the safe dose range?

Section 3.6

1. According to The Step Diet Book, the number of steps it takes for a 150 lb person to burn off the calories from eating a cheeseburger and drinking a 12 oz soda is 11,040. The number of steps it takes to burn off the calories from eating a cheeseburger is 690 more than twice the number of steps it takes to burn off the calories from drinking a 12 oz soda.

(a) Let x represent the number of steps necessary for a 150 lb person to burn off the calories from eating a cheeseburger and let y represent the number of steps necessary for a 150 lb person to burn off the calories from drinking a 12 oz soda. Finish the equation. .

(b) Solve the equation for y. Is y a function of x? Explain.

(c) Using that the number of steps it takes to burn off the calories from eating a cheeseburger is 690 more than twice the number of steps it takes to burn off the calories from drinking a 12 oz soda, write an equation giving the relationship between x and y.

(d) Use your answer from part (b) to rewrite your equation in (c) with only one variable.

(e) Determine the number of steps needed for a 150 lb person to burn off the calories from eating a cheeseburger and from drinking a 12 oz soda.

2. The amounts d (in billions of dollars) spent on prescription drugs in the United States from 1991 through 2005 (see figure) can be approximated by the model

where t represents the year, with t = 1 corresponding to 1991.

Find the amounts spent on prescription drugs in 1997, 2000, and 2004.

Section 3.7

1. The temperature of a patient after being given a fever-reducing drug is given by where F is the temperature in degrees Fahrenheit and t is the time in hours since the drug was administered.

(a) Is this relation a function?

(b) If you were just given the graph of the relation, how would you determine if it represented a function?

(c) Describe how the graph of F(t) is related to the graph of