Math 100 Workshop Linear Functions: Modeling and Applications DAY 3

Math 116A – Linear Functions: Modeling and Applications

IN ALL PROBLEMS YOUR ANSWERS MUST CONTAIN UNITS. THE GRAPHS, EXCEPT FOR THE SCATTER-DIAGRAM MUST BE DONE WITHOUT THE CALCULATOR DO NOT WASTE TIME WITH SCALES, A ROUGH SKETCH THAT MAKES SENSE IS OK LABEL X- AND Y-INTERCEPTS WHEN APPROPRIATE YOU MUST LABEL AXES WITH SYMBOLS, VARIABLE NAMES AND UNITS

1. Advertising Expenses The total amount of advertising expenses (in billions of dollars) in the United States from 1980 to 1989 is approximated using the linear model E = 8.1437 t + 52.982, 0 ≤ t ≤ 9 where E represents the advertising expenses (in billions of dollars) and t represents the year with t = 0 corresponding to 1980. a) What were the advertising expenses in the year 1983? b) When were the advertising expenses 118 billion dollars? c) What is the slope of this line? Interpret within the context of the problem. d) What is the y-intercept? Interpret within the context of the problem. e) Interpret in words the meaning of the ordered pair (4, 85.6) f) Sketch a graph. Label axes using variable-names and units. Label any important features. g) Explain in words the meaning of the x-intercept. Does it make sense?

2. Air Temperature and Altitude The relationship between the air temperature T (in ºF) and the altitude h (in feet above sea level) is approximately linear for 0 ≤ h ≤ 20,000 .If the temperature at sea level is 60 º, an increase of 5000 feet in altitude lowers the air temperature about 18 º. (a) Express T in terms of h, and sketch the graph. (b) Approximate the air temperature at an altitude of 15,000 feet. (c) Approximate the altitude at which the temperature is 0 º . What is the name of this point in the graph?

3. Nutrition. There are approximately 126 calories in a 2-ounce serving of lean hamburger and approximately 189 calories in a 3-ounce serving. a) Write a linear equation for the number of calories in lean hamburger in terms of the size of the serving. b) Use your equation to estimate the number of calories in a 5-ounce serving of lean hamburger. c) Interpret the ordered pair (10, 630)

4. Olympic Swimming The winning times in the women’s 400-meter freestyle swimming event in the Olympics from 1948 to 1988 are modeled by the equation y = 5.5 – 0.033t, 8 ≤ t ≤ 48 Where y represents the winning time in minutes and t represents the year with t = 0 corresponding to 1940. a) What was the winning time in the year 1980? b) When was the winning time 4.44 minutes? c) Give the value of the y-intercept, and interpret. Does it make sense? d) What is the value of the slope? Interpret. e) Find the x-intercept. Interpret. Does it make sense? f) Sketch the graph and label important features. 5. Childhood weight A baby weighs 10 pounds at birth, and three years later the child's weight is 30 pounds. Assume that childhood weight W (in pounds) is linearly related to age t (in years). (a) Express W in terms of t. (b) What is the weight of the child on his/her sixth birthday? (c) At what age will the child weigh 70 pounds? (d) Sketch the graph on the interval [0,12] (e) Interpret the x(t-)intercept. Does it make sense?

6. Loan repayment A college student receives an interest-free loan of $8250 from a relative. The student will repay $125 per month until the loan is paid off. (a) Express the amount L (in dollars) remaining to be paid in terms of time t (in months). (b) After how many months will the student owe $5000? (c) Find the x-intercept and interpret. (d) Sketch a graph and label important features.

7. Flying Lessons Flying lessons cost $645 for an 8 hour course and $1425 for a 20 hour course. Both prices include a fixed insurance fee. Express the cost, C, of flying lessons in terms of the length, h, of the course. What is the fixed insurance fee? Sketch a graph and label.

8. House appreciation Six years ago a house was purchased for $89,000. This year it was appraised at $125,000. Assume that the value V of the house after its purchase is a linear function of time t (in years). (a) Express V in terms of . (b) How many years after the purchase date was the house worth $103,000? (c) What is the meaning of the x-intercept? Does it make sense within the context of the problem?

9. Caramel Apple Sales A vendor has learned that, by pricing caramel apples at $1.25, sales will reach 133 caramel apples per day. Raising the price to $2.25 will cause the sales to fall to 81 caramel apples per day. Let y be the number of caramel apples the vendor sells at x dollars each. Write a linear equation that models the number of caramel apples sold per day when the price is x dollars each. What is the meaning of the x-intercept?

10. Sales Commissions A person applying for a sales position is offered alternative salary plans. Plan A: A base salary of $1,200 per month plus a commission of 4% of the gross sales per month Plan B: A base salary of $1,400 per month plus a commission of 3% of the gross sales for the month. a. For each plan, write an equation that expresses monthly earnings as a function of gross sales. b. Graph both equations on the same set of axes. Label. c. For what gross-sales values is plan B preferable? Selected Answers

1. a) 77.4 billion dollars, b) In 1988

2. a) T = (-9/2500) h + 60, b) T = 6 ºF, c) h = 16,667 ft.

3. y = 63 x, 315 cal.

4. a) 4.18 minutes, b) In 1972

5. a) W = (20/3) t + 10, b) 50 lb, c) 9 years old. d) the graph has endpoints at (0,10), and (12,90)

6. a) P = -125t +8250, b) 26 months, c) 66 months,

7. C = 65 h + 125; $125

8. a) V = 6000t + 89,000, b) 2.3years

9. y = -52x + 198

10. a) Plan A: y = 1200 + .04 x; Plan B: y = 1400 + 0.03x, c) for less than $20,000 The following problems do not have answers

1. Biology Biologists have observed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 120 chirps per minute at 70 °F and 168 chirps per minute at 80°F. a) Find the linear equation that relates the temperature T and the number of chirps per minute N. b) What is the slope? Interpret in context. c) What is the y-intercept? Interpret in context. c) If the crickets are chirping at 150 chirps per minute, estimate the temperature. d) How many chirps per minute at 90°F? e) Sketch a graph and label.

2) World record times for the men’s 400-meter run are listed in the table shown below:

Year Record time (seconds) 1900 47.8 1916 47.4 1928 47.0 1932 46.2 1941 46.0 1950 45.8 1960 44.9 1968 43.86 1988 43.29 1999 43.18

Let R represent the record time (in seconds) at t years since 1900

a) Use your calculator to sketch a scattergram of the data. Label axes using symbols, variable names, and units. Show a rough sketch above. Do not waste time with scales. b) Use the calculator to find the line of best fit. Write the equation using 3 decimal places. c) What is the y-intercept of this equation? What is the meaning within the context of the problem? d) What is the slope of this equation? What is the meaning within the context of the problem? e) What is the x-intercept of this equation? What is the meaning within the context of the problem? Comment on model breakdown. f) Find t if R(t) = 42 seconds. Interpret your results in words. g) Find R(36) and interpret within the context of the problem.

3. Olympic Swimming The winning times in the women’s 400-meter freestyle swimming event in the Olympics from 1948 to 1988 are modeled by the equation y = 5.5 – 0.033t, 8 ≤ t ≤ 48 Where y represents the winning time in minutes and t represents the year with t = 0 corresponding to 1940.

a) What was the winning time in the year 1980? b) When was the winning time 4.44 minutes? c) Give the value of the y-intercept, and interpret in context. d) What is the value of the slope? Interpret in context. e) Find the x-intercept. Interpret. Does it make sense? Comment on model breakdown f) Sketch the graph and label. 4. Gas mileage

The gas mileage, m, of a compact car is a linear function of the speed, s, at which the car is driven, for For example, from the graph we see that the gas mileage for the compact car is 45 miles per gallon if the car is driven at a speed of

Find the average rate of change in gas mileage between speeds of 40 mph and 60 mph. Find the average rate of change in gas mileage between speeds of 50 mph and 70 mph. Find the average rate of change in gas mileage between speeds of 70 mph and 90 mph.