Mathematical Models 1.1 Includes…geometric formulas, regression analysis, solving equations in 1 variable

Definitions A mathematical model is a mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior One type of mathematical model: numerical model, where numbers (or data) are analyzed to gain insights into phenomena

period during which the Table 1.1 The Minimum Hourly Wage Year Min. Hourly Purchasing Power Wage in 2001 Dollars 1940 0.30 3.68 1945 0.30 2.88 1950 0.75 5.43 1955 0.75 4.79 1960 1.00 5.82 1965 1.25 6.84 1970 1.60 7.23 1975 2.10 6.88 1980 3.10 6.80 1985 3.35 5.48 1990 3.35 4.57 1995 4.25 4.94 2000 5.15 5.34 In what year did a minimum-wage worker have the greatest purchasing power? In 1970 What was the longest period during which the minimum wage did not increase? From 1940-1945, 1950-1955, and 1985-1990

nearly twice as much as a Table 1.1 The Minimum Hourly Wage Year Min. Hourly Purchasing Power Wage in 2001 Dollars 1940 0.30 3.68 1945 0.30 2.88 1950 0.75 5.43 1955 0.75 4.79 1960 1.00 5.82 1965 1.25 6.84 1970 1.60 7.23 1975 2.10 6.88 1980 3.10 6.80 1985 3.35 5.48 1990 3.35 4.57 1995 4.25 4.94 2000 5.15 5.34 A worker making min. wage in 1980 was earning nearly twice as much as a worker making min. wage in 1970  so why was there pressure to once again raise the min. wage? Purchasing power actually dropped by $0.43 during that period (inflation)

Definitions Another type of mathematical model: Algebraic Model – uses formulas to relate variable quantities associated with the phenomena being studied (Benefit: can generate numerical values of unknown quantities using known quantities)

Guided Practice At Dominos, a small (10” diameter) cheese pizza costs $4.00, while a large (14” diameter) cheese pizza costs $8.99. Assuming that both pizzas are the same thickness, which is the better value? Calculate areas per dollar cost: Small Pizza Large Pizza 2 2 Small: 19.635 in /$, Large: 17.123 in /$ The small pizza is the better value!!!

Regression Analysis: Graphical Model – visual representation of a numerical or algebraic model that gives insight into the relationships between variable quantities Regression Analysis: The process of analyzing data by creating a scatter plot, critiquing the data’s appearance (linear, parabolic, cubic, etc.), choosing the appropriate model, finding the line of best fit, making predictions about the data.

A Good Example: Galileo gathered data on a ball rolling down an inclined plane: Elapsed Time (seconds) 1 2 3 4 5 6 7 8 Distance Traveled (in) .75 6.75 12 18.75 27 36.75 48 1. Create a scatter plot of these data 2. Derive an algebraic model to fit these data 2 d = 0.75t 3. Graph this function on top of your scatter plot

A Good Example: d = 168.75 in t = 9.092 sec Galileo gathered data on a ball rolling down an inclined plane: Elapsed Time (seconds) 1 2 3 4 5 6 7 8 Distance Traveled (in) .75 6.75 12 18.75 27 36.75 48 4. How far will the ball have traveled after 15 seconds? d = 168.75 in 5. How long will it take the ball to travel 62 inches? t = 9.092 sec

Terminology: If a is a real number that solves the equation f(x) = 0, then these three statements are equivalent: 1. The number a is a root (or solution) of the equation f(x) = 0. 2. The number a is a zero of y = f(x). 3. The number a is an x-intercept of the graph of y = f(x). (sometimes the point (a, 0) is referred to as an x-intercept)

More Practice Problems… 3 2 Find all real numbers x for which 6x = 11x + 10x 5 2 x = 0 or x = or x = – 2 3 We just used the Zero Product Property: A product of real numbers is zero if and only if at least one of the factors in the product is zero.

Guided Practice Solve the equation algebraically and graphically.

Guided Practice Solve the equation algebraically and graphically. Use the quadratic formula: Check for extraneous solutions!!!

Practice 3 Find all x-intercepts of y = 2x – 5 Solving algebraically??? (there are no x-intercepts) Solving graphically???

Try another problem: x = –9, 5, 5.1 Solve graphically: x – 1.1x – 65.4x + 229.5 = 0 3 2 x = –9, 5, 5.1 This is an example of hidden behavior – when an inaccurate viewing window obscures details of a graph

Now, on to some basic problem solving… The engineers at an auto manufacturer pay students $0.08 per mile plus $25 per day to road test their new vehicles. 1. Derive an algebraic model for the students’ pay. p = 0.08x + 25 (where p = students’ pay, and x = miles driven)

Now, on to some basic problem solving… The engineers at an auto manufacturer pay students $0.08 per mile plus $25 per day to road test their new vehicles. 2. How much did the auto manufacturer pay Sally to drive 440 miles in one day? $60.20 3. John earned $93 test-driving a new car in one day. How far did he drive? 850 miles

Now, on to some basic problem solving… A math student’s grade is determined by weights, with home- work counting 20% and quizzes/tests counting 80%. Derive an algebraic model for the student’s grade, given that homework and quizzes/tests are graded by points. Total HW pts. Total Q/T pts. g = 0.2 + 0.8 Poss. HW pts. Poss. Q/T pts.

Now, on to some basic problem solving… A math student’s grade is determined by weights, with home- work counting 20% and quizzes/tests counting 80%. 2. What is Wolfgang’s grade if he has earned 26 out of a possible 28 homework points, quiz grades of 22/25 and 13/15, and a test grade of 41/50? Grade = 86.127%

Now, on to some basic problem solving… A math student’s grade is determined by weights, with home- work counting 20% and quizzes/tests counting 80%. 3. If Jan has perfect homework scores and quiz scores of 20/25 and 14/15, what does she need to earn on the upcoming 50-point test in order to have an overall 90% grade average? At least 44.75 points

Now, on to some basic problem solving… Solve the given equation graphically by converting it to an equivalent equation with 0 on the right-hand side and then finding the x-intercepts. 1. 2. Homework: Odds p.76 #7-11, 15, 17, 21, 29-47