1. Math 101. Math 101 Website: richard/math101 Section 2 Website: richard/math101/fall06

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1 1. Math 101 Lines and Slope Professor Richard Blecksmith Dept. of Mathematical Sciences Northern Illinois University Math 101 Website: richard/math101 Section 2 Website: richard/math101/fall06 2. Construction Problem A contractor builds two types of homes: the standard model and the deluxe model. The standard model requires one lot, $12,000 capital, 150 labor-days to build, and is sold for a profit of $2400. The deluxe model requires one lot, $32,000 capital, 200 labor-days to build, and is sold for a profit of $3400. The contractor has 150 lots. The bank is willing to loan him $2,880,000 for the project and he has a maximum labor force available of 24,000 labor-days. How many houses should he build to realize the greatest profit? Slope Intercept Form y = mx + b m is the slope b is the y-intercept 3. Two Forms of a Line 1

2 2 Standard Form ax + by = c 4. Point Slope Formula If we know (1) the slope m of a line and (2) a point P = (x 1, y 1 ) on the line then we can use The Point Slope Formula y y 1 = m(x x 1 ) to find the equation of the line 5. Point Slope Example Problem. Find the equation of the line that passes through the point (2,3) and has the slope m = 1 2. Solution. In the Point Slope Formula (1) the slope m = 1 2 and (2) the point P = (2, 3) on the line Plugging these into the Point Slope Formula y y 1 = m(x x 1 ) yields y 3 = 1 2 (x 2) y 3 = 1 2 x 1 y = 1 2 x Slope Problem: How to find the slope of a line: If you know two points P = (x 1, y 1 ) and Q = (x 2, y 2 ) Subtract: y 2 y 1

3 3 Subtract: x 2 x 1 Now divide: slope = y 2 y 1 x 2 x 1 Example: Find the slope of the line through the points P = (1, 2) and Q = (3, 8) (1,2) 7. Graph (3,8) slope = = 6 2 = 3 8. Line thru Two Points Problem. Find the equation of the line passing through the points (1, 2) and (3, 8) Solution. Once we know the slope m = = 6 2 = 3 we can plug m = 3 and P = (1, 2) into the Point Slope Formula

4 4 y y 1 = m(x x 1 ) to get y 2 = 3(x 1) y 2 = 3x 3 y = 3x 1 9. The Distance Formula The distance between two points (x 1, y 1 ) and (x 2, y 2 ) is given by The Distance Formula distance = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 The distance between (1, 2) and (3, 8) is distance = (3 1) 2 + (8 2) 2 = = = Converting Celcius to Fahrenheit Note that Celsius and Centigrade are interchangable They refer to the metric system measurement of temperature The problem in converting temperature is that the 0 points are different in the two systems 0 meters = 0 inches Both signify no length 0 square meters = 0 square yards Both signify no area But 0 degrees centigrade = 32 degrees Fahrenheit

5 5 the freezing point of water 11. Celcius to Fahrenheit Cont d You cannot just multiply centigrade temperature by a conversion factor to get the Fahrenheit temperature You must adjust for the fact that the zero point are different The conversion is represented by a line in the plane The x values are degrees Centigrade The y values are degrees Fahrenheit 12. Celcius to Fahrenheit Cont d We know two points on this line. Freezing point of water 0 Centigrade = 32 Fahrenheit So (0, 32) is a point on the line Boiling point of water 100 Centigrade = 212 Fahrenheit So (100, 212) is another point on the line

6 6 13. Graph of the Line 212 (100,212) Degrees Fahrenheit (0,32) Degrees Centigrade Finding the Equation The slope of this line is m = = = 9 5 The equation for our line is F = 9 5 C + b To determine the constant b, Plug in the point C = 0, F = 32:

7 7 32 = b So b = 32 and F = 9 5 C + 32 F = 9 5 C Converting Temperatures You are visiting Paris, France, and the a sign says the temperature is 16 Since France uses the metric system, you know this temperature is in Centigrade What s the temperature in Fahrenheit? Use the Formula F = = 60.8 or about 61 degrees. 16. A Practical Rule of Thumb The problem with this method is No one remembers the formula. The fractions are hard to compute in your head. Here s an easier method: Use the formula F = 2C + 30 For the previous problem, if C = 16, then F = = 62 This answer is very close to the correct answer of 60.8

8 8 17. Comparing the Lines 212 (100,212) Degrees Fahrenheit F = 2C + 30 F = 9 5 C + 32 (0,32) Degrees Centigrade Zooming In to Normal Temps Degrees Fahrenheit (0,32) F =2C+30 F = 9 5 C Degrees Centigrade

9 9 19. Conclusion As can be seen from the graph, for temperatures between 0 and 30 Centigrade, the second formula F = 2C + 30 is accurate to within 2 degrees. 20. Linear Versus Nonlinear Many functions are linear, that is, their graphs are straight lines. Example: You work at an hourly wage of $6.50. Then your weekly salary is linear, as a function of the number of hours worked: On your first hour, you make $6.50. On your second hour, you make $6.50. On your 19th hour, you make $6.50. In general, your salary is Salary = 6.5 hours worked 21. Some Non-Linear Functions The interest you pay on your mortgage is not linear. It is very large at the beginning of your loan and tapers off toward the end. The Dow Jones Average is not linear. It fluctuates from day to day. When you drive a car, the distance you travel is not linear (with respect to time) unless you are using cruise control The temperature in DeKalb is not linear.

10 PT Cruiser Example: Road & Track Magazine tested Chrysler s PT Cruiser. They needed 131 feet to stop the car from a speed of 60 mph. They needed 232 feet to stop the car from a speed of 80 mph. Increasing the speed by 20 mph almost doubles the distance to stop. This is not linear behavior. Moral: Not all functions are linear 23. Sir Issac Newton Sir Issac Newton: inventor of Calculus (and a popular figfilled cookie): Stopping distance increases with the square of the speed. Stopping Distance = K Speed 2 where K is the (braking) constant. 24. Stopping a PT Cruiser Plugging in Distance = 131 and Speed = 60 allows us to solve for K: 131 = K 60 2 Solve for K by dividing by 60 2 : K = =.0364 Stopping Distance =.0364 Speed 2

11 Newton s Prediction Stopping Distance =.0364 Speed 2 This equation predicts that when Speed = 80, the stopping distance will be Stopping Distance = = Compare this will the actual value found by Road & Track s test drivers: Stopping distance at 80 mph was 232 feet. Way to go, Newton. Not bad for a guy who predated the automobile by 200 years. 26. Stopping Distance Table Speed Stopping Distance Speed Stopping Distance Stopping Distance Graph Stopping Distance vs. Speed

12 Driving Considerations If you increase your speed by 5 mph, it requires an additional 4.6 feet to stop if you are going 10 to 15 mph. It requires an additional 30 feet to stop if you are going 80 to 85 mph. This is not a linear function. It take four times longer to stop from 60 mph than from 30 mph. It take nine times longer to stop from 90 mph than from 30 mph. Moral: Don t tailgate on the interstates.