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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics. §3.2a System Applications. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. MTH 55. 3.1. Review §. Any QUESTIONS About §’s3.1 → Systems of Linear Equations Any QUESTIONS About HomeWork §’s3.1 → HW-08. System Methods Compared.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Chabot Mathematics §3.2a SystemApplications Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. MTH 55 3.1 Review § • Any QUESTIONS About • §’s3.1 → Systems of Linear Equations • Any QUESTIONS About HomeWork • §’s3.1 → HW-08

  3. System Methods Compared • We now have three distinctly different ways to solve a system. Each method has strengths & weaknesses

  4. Solving Application Problems • Read the problem as many times as needed to understand it thoroughly. Pay close attention to the questions asked to help identify the quantity the variable(s) should represent. In other Words, FAMILIARIZE yourself with the intent of the problem • Often times performing a GUESS & CHECK operation facilitates this Familiarization step

  5. Solving Application Problems • Assign a variable or variables to represent the quantity you are looking for, and, when necessary, express all other unknown quantities in terms of this variable. That is, Use at LET statement to clearly state the MEANING of all variables • Frequently, it is helpful to draw a diagram to illustrate the problem or to set up a table to organize the information

  6. Solving Application Problems • Write an equation or equations that describe(s) the situation. That is, TRANSLATE the words into mathematical Equations • Solve the equation(s); i.e., CARRY OUT the mathematical operations to solve for the assigned Variables • CHECK the answer against the description of the original problem (not just the equation solved in step 4)

  7. Solving Application Problems • Answer the question asked in the problem. That is, make at STATEMENT in words that clearly addressed the original question posed in the problem description

  8. Example  Problem Solving • Two angles are supplementary. One of the angles is 20° larger than three times the other. Find the two angles • Familarize • Recall that two angles are supplementary if the sum of their measures is 180°. We could try and guess, but instead let’s make a drawing and translate. Let x and y represent the measures of the two angles

  9. Example  Problem Solving • Familarizewith Diagram x y • Translate • Since the angles are supplementary, one equation is x + y = 180 (1) • The second sentence can be rephrased and translated as follows:

  10. Example  Problem Solving • Translating • Rewording and Translating One angle is 20 more than three times the other y = 20 + 3x (2) • We now have a system of two equations and two unknowns.

  11. Example  Problem Solving • Carry Out • Sub x = 40° in Eqn-1

  12. Example  Problem Solving • Check • If one angle is 40° and the other is 140°, then the sum of the measures is 180°. Thus the angles are supplementary. If 20° is added to three times the smaller angle, we have 3(40°) + 20° = 140°, which is the measure of the other angle. The numbers check. • STATE • One angle measures 40° and the other measures 140°

  13. Elimination Applications • Total-Value Problems • Mixture Problems

  14. Total Value Problems • EXAMPLE: Lupe sells concessions at a local sporting event. • In one hour, she sells 72 drinks. The drink sizes are • small, which sells for $2 each • large, which sells for $3 each. • If her total sales revenue was $190, how many of each size did she sell?

  15. Example  Drinks Sold • Familiarize. • Suppose (i.e., GUESS) that of the 72 drinks, 20 where small and 52 were large. • The 72 drinks would then amount to a total of 20($2) + 52($3) = $196. • Although our guess is incorrect (but close), checking the guess has familiarized us with the problem.

  16. Example  Drinks Sold • Familiarize – LET: • s = the number of small drinks and • l = the number of large drinks • Translate. • Since a total of 72 drinks were sold, we must have s + l = 72. • To find a second equation, we reword some information and focus on the income from the drinks sold

  17. $3 $2 Example  Drinks Sold • Translate. • Translating & Rewording Income fromsmall drinks Income fromlarge drinks Totals $190 Plus $2s $3l = + $190 • Thus we haveConstructedthe System

  18. Example  Drinks Sold • Carry Out Solve (1) for l Use (3) to Sub for l in (2) Use Distributive Law Combine Terms Simplify to find s • Sub s = 26 in (3) to Find l

  19. Example  Drinks Sold • Check: If Lupe sold 26 small and 46 large drinks, she would have sold 72 drinks, for a total of: 26($2) + 46($3) = $52 + $138 = $190 • State: Lupe sold • 26 small drinks • 46 large drinks

  20. Problem-Solving TIP • When solving a problem, see if it is patterned or modeled after a problem that you have already solved.

  21. Example  Problem Solving • A cookware consultant sells two sizes of pizza stones. The circular stone sells for $26 and the rectangular one sells for $34. In one month she sold 37 stones. If she made a total of $1138 from the sale of the pizza stones, how many of each size did she sell? Pizza Stone

  22. Example  Pizza Stones • Familiarize –When faced with a new problem, it is often useful to compare it to a similar problem that you have already solved. Here instead of $2 and $3 drinks, we are counting $26 & $34 pizza stones. So LET: • c = the no. of circular stones • r = the no. of rectangular stones

  23. c + r = 37 26c + 34r = 1138 Example  Pizza Stones cont.2 • Translate –Since a total of 37 stones were sold, we have: c + r = 37 • Tabulating the Data Can be Useful Circular Rectangular Total Cost per pan $26 $34 Number of pans c r 37 Money Paid 26c 34r $1138

  24. Example  Pizza Stones • Translate –We have translated to a system of equations: c + r = 37 (1) 26c + 34r = 1138 (2) • Carry Out Multiply Eqn(1) by −26 • Add Eqns (2)&(3):

  25. Example  Pizza Stones cont.4 • Carry Out – Solve for r • Find c using Eqn (1): Check: If r = 22 and c = 15, a total of 37 stones were sold. The amount paid was 22($34) + 15($26) = $1138  State: The consultant sold 15 Circular and 22 rectangular pizza stones

  26. Example  Mixture Problem • A Chemical Engineer wishes to mix a reagent that is 30% acid and another reagent that is 50% acid. • How many liters of each should be mixed to get 20 L of a solution that is 35% acid?

  27. t liters + f liters = 20 liters 30% acid 50% acid 35% acid Example  Acid Mixxing • Familiarize. Make a drawing and then make a guess to gain familiarity with the problem • The Diagram

  28. Example  Acid Mixxing • To familiarize ourselves with this problem, guess that 10 liters of each are mixed. The resulting mixture will be the right size but we need to check the Pure-Acid Content: • Our 10L guess produced 8L of pure-acid in the mix, but we need 0.35(20) = 7L of pure-acid in the mix

  29. Example  Acid Mixxing • Translate: • LET • t = the number of liters of the 30% soln • f = the number of liters of the 50% soln • Next Tabulate the calculation of the amount of pure-acid in each of the mixture components

  30. Example  Acid Mixxing • Pure-Acid Calculation Table • The Table Reveals a System of Equations

  31. Example  Acid Mixxing • Carry Out: Solve Eqn System • Eliminate f by multiplying both sides of equation (1) by −0.5 and adding them to the corresponding sides of equation (2): –0.50t – 0.50f = –10 0.30t + 0.50f = 7 –0.20t = –3 t = 15.

  32. Example  Acid Mixxing • To find f, we substitute 15 for t in equation (1) and then solve for f: 15+ f = 20 f = 5 • Obtain soln (15, 5), or t = 15 and f = 5 • Check: Recall t is the of liters of 30% soln and f is the of liters of 50% soln  Number of liters: t + f = 15 + 5 = 20 Amount of Acid: 0.30t + 0.50f = 0.30(15)+ 0.50(5) = 7

  33. Example  Wage Rate • Ethan and Ian are twins. They have decided to save all of the money they earn at their part-time jobs to buy a car to share at college. One week, Ethan worked 8 hours and Ian worked 14 hours. Together they saved $256. The next week, Ethan worked 12 hours and Ian worked 16 hours and they earned $324. • How much does each twin make per hour? • i.e.; What are the Wage RATES

  34. Example  Wage Rate • In This Case LET: • E≡ Ethan’s Wage Rate ($/hr) • I≡ Ian’s Wage Rate ($/hr) • Translate: Ethan worked 8 hours and Ian worked 14 hours. Together they saved $256 8∙{Ethan’s Rt} plus 14∙{Ian’s Rt} is $256

  35. Example  Wage Rate • Translate: Ethan worked 12 hours and Ian worked 16 hours and they earned $324. 12∙{Ethan’s Rt} plus 16∙{Ian’s Rt} is $324 • Now have 2-Eqn System

  36. Example  Wage Rate • Carry Out

  37. Example  Wage Rate • Carry Out • Sub I = 12 into 1st eqn to Find E • State Answer • Ethan Earns $11 per hour • Ian Earns $12 per hour

  38. l w Example  Geometry • The perimeter of a fence around the children’s section of the community park is 268 feet. The length is 34 feet longer than the width. Find the dimensions of the park. • Familiarize: Draw a Diagram and LET: • l≡ the length • w≡ the width

  39. Example  Geometry • Translate. • The Perimeter is 2l + 2w. The perimeter is 268 feet. 2l + 2w = 268 • The length is 34 ft more than the width l = 34 + w

  40. Example  Geometry • Now have a system of two equations and two unknowns. • 2l + 2w = 268 (1) • l = 34 + w (2) • Solve for wusingSubstitutionMethod

  41. Example  Geometry • Sub 50 for w in one of the Orignal Eqns l = 34 + w = 34 + 50 = 84 feet Check: If the length is 84 and the width is 50, then the length is 34 feet more than the width, and the perimeter is: 2(84) + 2(50), or 268 feet State: The width is 50 feet and the length is 84 feet

  42. Example  Mixture Problem • A coffee shop is considering a new mixture of coffee beans. It will be created with Italian Roast beans costing $9.95 per pound and the Venezuelan Blend beans costing $11.25 per pound. The types will be mixed to form a 60-lb batch that sells for $10.50 per pound. • How many pounds of each type of bean should go into the blend?

  43. Example  Coffee Beans • Familiarize –This problem is similar to one of the previous examples. • Instead of pizza stones we have coffee beans • We have two different prices per pound. • Instead of knowing the total amount paid, we know the weight and price per pound of the new blend being made. • LET: • i≡ no. lbs of Italian roast and • v≡ no. lbs of Venezuelan blend

  44. i + v = 60 9.95i + 11.25v = 630 Example  Coffee Beans • Translate –Since a 60-lb batch is being made, we have i + v = 60. • Present the information in a table.

  45. Example  Coffee Beans cont.4 • Translate -We have translated to a system of equations • Carry Out -When equation (1) is solved for v, we have: v = 60 −i. • We then substitute for v in equation (2).

  46. Example  Coffee Beans cont.5 • Carry Out -Find v using v = 60 −i. • Check -If 34.6 lb of Italian Roast and 25.4 lb of Venezuelan Blend are mixed, a 60-lb blend will result. • The value of 34.6 lb of Italian beans is 34.6•($9.95), or $344.27. • The value of 25.4 lb of Venezuelan Blend is 25.4•($11.25), or $285.75,

  47. Example  Coffee Beans cont.6 • Check –cont. • so the value of the blend is [$344.27 + $285.75] = $630.02. • A 60-lb blend priced at $10.50 a pound is also worth $630, so our answer checks • State–The blend should be made from • 34.6 pounds of Italian Roast beans • 25.4 pounds of Venezuelan Blend beans

  48. Simple Interest • If a principal of P dollars is borrowed for a period of t years with interest rate r (expressed as a decimal) computed yearly, then the total interest paid at the end of t years is • Interest computed with this formula is called simpleinterest. When interest is computed yearly, the rate r is called an annual interest rate

  49. Example  Simple Interest • Ms. Jeung invests a total of $10,000 in Bonds from blue-chip and technology Companies. At the end of a year, the blue-chips returned 12% and the technology stocks returned 8% on the original investments. • How much was invested in each type of Bond if the total interest earned was $1060?

  50. Example  Simple Interest • Familiarize: • We are asked to find two amounts: • that invested in blue-chip Bonds • that invested in technology Bonds. • If we know how much was invested in blue-chip Bonds, then we know that the rest of the $10,000 was invested in technology Bonds.

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