Review Document MTH-2101-3 Created by Martine Blais Commission scolaire des Premières-Seigneuries Mai 2010 Translated by Marie-Hélène Lebeault November 2017 Producing an algebraic model 1. How do you go about producing an algebraic model? What steps should you follow? 2. What should you always identify when producing an algebraic model? 1
A few algebraic models seen in this class 3. Which algebraic model lets you calculate the weekly salary of an employee who earns $12 per hour? 4. Which algebraic model lets you calculate the weekly salary of a waitress who earns $9 per hour plus an average of 15% in commissions? 5. Which algebraic model lets you establish a relation between the distance, speed, and time? a. b. c. 2
6. Which algebraic model lets you calculate the cost of gas, taking into account gas consumption and the distance traveled, if gas is sold at $1.03 per liter? Image Office 7. Which algebraic model lets you calculate the weekly salary of a waitress who earns $9 per hour plus a $25 commission on each sale? 8. Which algebraic model lets you calculate the cost of buying a car if there is a 10% rebate and that you must pay $500 fee for preparation? 9. Which algebraic model lets you calculate the cost of a taxi ride, if the base rate is $3.30, to which you add $1.60 per km and $0.60 per minute of waiting? 3
Proportionality Situations Variables can be directly proportional, inversely proportional or nonproportional. Directly Proportional Variables 10. What can you say about directly proportional variables? 11. When one of the directly proportional variables is multiplied or divided by a number, what happens to the other variable? 12. Provide an example of directly proportional variables. 13. How can you tell if variables are directly proportional when looking at an algebraic model? 4
14. How can you tell if variables are directly proportional when looking at a real-life situation? Inversely Proportional Variables 15. What can you say about inversely proportional variables? 16. When one of the inversely proportional variables is multiplied or divided by a number, what happens to the other variable? 17. Provide an example of directly proportional variables. 5
18. How can you tell if variables are inversely proportional when looking at an algebraic model? 19. How can you tell if variables are inversely proportional when looking at a real-life situation? Variables that are not proportional 20. What can you say about variables that are not proportional? 21. When one of the non-proportional variables is multiplied or divided by a number, what happens to the other variable? 6
22. Provide an example of non-proportional variables. 23. How can you tell if variables are non-proportional when looking at an algebraic model? 24. How can you tell if variables are non-proportional when looking at a reallife situation? 7
Types of Algebraic Models Name the type of algebraic model that can be produced on the following situations and produce the corresponding algebraic model. Answer Choices: Linear Equation Proportion (directly proportional variables) Proportion (inversely proportional variables) 25. A plumber charges $80 per call, $75 per hour. Image Office 26. Four worker put up a support wall in 3 days. If there were six workers, how long would it take? 8
27. A salesman earns $12 per hour plus an 8% commission on all sales. Image Office 28. When borrowing $1,000, we pay $70 in interest. How much interest would be paid if we borrow $700? 9
29. If it takes 5 hours to drive to the beach at 95 km/h, how long would it take if we drove at 100 km/h? 30. According to a survey, 80% of students at an Adult Education Center have a cell phone. If 300 students at the center have a cell phone, how many don t have one? Image Office 10
Solving Equations Solving an equation means finding the value of a variable so you can verify the equation. Here s an example: 1 7x 3 x + = + 2 3 36x + 12 6 12 3 4 28x = + 12 36 x + 6 = 28x + 9 36x - 28x = 9-6 8 x = 3 8 x 3 = 8 8 x = 3 8 9 12 Here s how to proceed when there are parentheses in the equation: 3 (2x - 7) = 5 - (-x + 4) 6x - 21 = 5 + x - 4 6 x - 21 = 1+ x 6 x - x = 1+ 21 5 x = 22 5 x = 5 22 5 x = 4,4 11
When the variable we are looking for is squared, solving the equation ends with a square root. Here is how to proceed: 2 2 a + b = a a c 2 2 2 2 + 4 = 7 2 + 16 = 49 a 2 = 49-16 a 2 = 33 a = 33 a = 5,74 When an equation is proportional, we use the Fundamental Law of proportions to solve it. Here s an example: 2x + 5 x = 6 2 2 (2x + 5) = 6 x 4 x + 10 = 6x 4x - 6x = -10-2x = -10-2x - 2 = -10-2 x = 5 12
Using the algebraic models you produced in the previous number, solve the equations to answer the following questions. 31. A plumber charges $80 per call, $75 per hour. How many hours did he work if the bill came to $492.50? 32. Four worker put up a support wall in 3 days. If there were six workers, how long would it take? 13
33. A salesman earns $12 per hour plus an 8% commission on all sales. If he earned $620 for a 35-hour work week, how much where his sales for that week? 34. When borrowing $1,000, we pay $70 in interest. How much interest would be paid if we borrow $700? 14
35. If it takes 5 hours to drive to the beach at 95 km/h, how long would it take if we drove at 100 km/h? Geometry Situations Image Office 36. What is a geometry situation? 37. In a situation involving geometric concepts, what steps should you take to make your work easier faire? 15
Geometry Concepts 38. Name the geometric concept Finding the outline of an object? Covering a floor with tiles? Determining how much liquid is in a glass? Painting the walls of a room? Determining how much cardboard is required to make a box? 39. How do you calculate the volume of a solid when you know the area of its base? The Pythagorean Theorem In a right triangle, the sum of the squares of the sides on the sides of the right angle is equal to the square of the hypotenuse. This is how you write the Pythagorean Theorem: 2 2 a + b = where a and b are the measures of the sides on either side of the right angle and c is the measure of the hypotenuse. c 2 40. We use the Pythagorean Theorem to find the of a side in a right triangle as well as 16
41. Name the flat figures studied in class, draw a sketch for each, and write the formulas for perimeter (circumference for circle) and area. Name Sketch Perimeter Area 17
42. Name the solids studied in class, draw a sketch for each, and write the formulas for volume, lateral area and total area. Name Sketch Volume Lateral Area Total Area 18
Units of Measure 43. Which measuring units are used to calculate The perimeter of a figure? The area of a figure, the lateral area or the total area of a solid? The volume or the capacity of a solid? 44. How do you convert feet into metres? 45. Convert 2,3m into centimetres. 19
46. Convert 4,5 km into metres. 47. Convert 18 inches into centimetres. Using Formulas 48. How do you go about solving a problem that requires the use of a formula? What steps and which processes can help you? 49. What should you check before using a formula? 20
Solve the following problems using an algebraic model: 50. What is the diameter of a cylindrical silo 10 m high that can hold 282.6 m 3 of fodder? If a silo of the same volume had a base of 25 m 2, what would be its height? Image Office 21
51. To convert Celsius into degrees Fahrenheit, we use the following C F - 32 formula: =. Use this formula to convert 72 F into C. Then, 5 9 convert -10 C into F. 22
52. Julie wants to make a rectangular patio. She plants stakes that indicate where the corners of the patio will be and she walks around with a 14-meter rope. The longest side of the future patio is 1 m longer than the shorter side. Using an algebraic model, determine the measure of the sides of the figure. 23
53. To enclose his field, Max used 80 m of fence. If the ground is three times wider than deep, what is the area of the land? 24
54. A swimming pool has a diameter of 7 m. After 2 hours of filling with a hose, the water reaches a height of 30 cm. How many liters of water does it contain at this time? How long will the filling take in all if the water is to reach 1.2 m in height? If two hoses with the same flow rate were used for filling, how long would the filling take in total? Image Office 25