1 / 40

COLLEGE ALGEBRA

This book covers various problem-solving techniques using linear equations, including geometry, motion, and mixture problems. It provides step-by-step instructions and examples to help students understand and solve applied problems.

chuckm
Download Presentation

COLLEGE ALGEBRA

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER

  2. Applications and Modeling with Lienar Equations 1.2 Solving Applied Problems Geometry Problems Motion Problems Mixture Problems Modeling with Linear Equations

  3. SOLVING AN APPLIED PROBLEM Step 1 Read the problem carefully until you understand what is given and what is to be found. Step 2 Assign a variable to represent the unknown value, using diagrams or table as needed. Write down what the variable represents. If necessary, express any other unknown values in terms of the variable.

  4. SOLVING AN APPLIED PROBLEM Step 3 Write an equation using the variable expression(s). Step 4 Solve the equation. Step 5 State the answer to the problem. Does it seem reasonable? Step 6 Check the answer in the words of the original problem.

  5. FINDING THE DIMENSIONS OF A SQUARE EXAMPLE 1 If the length of each side of a square is increased by 3 cm, the perimeter of the new square is 40 cm more than twice the length of the original square. Find the dimensions of the original square

  6. FINDING THE DIMENSIONS OF A SQUARE EXAMPLE 1 Solution Step 1 Read the problem. We must find the length of the original square. x + 3 x x x + 3

  7. FINDING THE DIMENSIONS OF A SQUARE EXAMPLE 1 Step 2 Assign a variable. Since the length of a side of the original square is to be found, let the variable equal this length. x Original square x The length of a side of the new square is 3 cm more than the length of the old square, so… Square increased by 3 x + 3 x + 3

  8. FINDING THE DIMENSIONS OF A SQUARE EXAMPLE 1 Step 2 Assign a variable. Since the length of a side of the original square is to be found, let the variable equal this length. Write a variable expression for the perimeter of the new square. The perimeter of a square is 4 times the length of the side.

  9. FINDING THE DIMENSIONS OF A SQUARE EXAMPLE 1 Step 3 Write an equation. twice the length of the side of the original square. The new perimeter more than is 40

  10. FINDING THE DIMENSIONS OF A SQUARE EXAMPLE 1 Step 4 Solve the equation. Distributive property Subtract 2x and 12. Divide by 2.

  11. FINDING THE DIMENSIONS OF A SQUARE EXAMPLE 1 Step 5 State the answer. Each side of the original square measures 14 cm.

  12. FINDING THE DIMENSIONS OF A SQUARE EXAMPLE 1 • Step 6 Check. Go back to the words of the original problem to see that all necessary conditions are satisfied. • length of side of new square: 14 + 3 = 17 cm • perimeter would be 4(17)= 68 cm • twice the length of side of original square is 2(14) = 28 cm • since 40 + 28 = 68, the answer checks

  13. Motion Problems Problem Solving In a motion problem, the components, distance, rate, and time are denoted by the letters d, r, and t respectively. (The rate is also called the speed or velocity. Here, rate is considered to be constant.) and its related forms and

  14. SOLVING A MOTION PROBLEM Example 2 Maria and Eduardo are traveling to a business conference. The trip takes 2 hrs for Maria and 2.5 hours for Eduardo, since he lives 40 miles farther away. Eduardo travels 5 mph faster than Maria. Find their average rates.

  15. SOLVING A MOTION PROBLEM Example 2 Solution Step 1 Read the problem. We must find both Maria’s and Eduardo’s average rates.

  16. SOLVING A MOTION PROBLEM Example 2 Step 2 Assign a variable. Let x = Maria’s rate. Then let x + 5 equal Eduardo’s rate. Summarize using a table. Use d = rt.

  17. SOLVING A MOTION PROBLEM Example 2 Step 3 Write an equation. Use the fact that Eduardo’s distance traveled exceeds Maria’s distance by 40 mi. Eduardo’s distance 40 more than Maria’s. is

  18. SOLVING A MOTION PROBLEM Example 2 Step 4 Solve. Eduardo’s distance 40 more than Maria’s. is Distributive property Subtract 2x; subtract 12.5 Divide by .5

  19. SOLVING A MOTION PROBLEM Example 2 Step 5 State the answer. Maria’s rate of travel is 55 mph, and Eduardo’s rate is… 55 + 5 = 60 mph. Step 6 Check. The diagram shows that the conditions of the problem are satisfied. Distance traveled by Maria: 2(55) = 110 mi. Distance traveled by Eduardo: 2.5(60) = 150 mi. 150 – 110 = 40

  20. Mixture Problems Problem Solving In mixture problems, the rate (percent) of concentration is multiplied by the quantity to get the amount of pure substance present. The concentration of the final mixture must be between the concentrations of the two solutions making up the mixture.

  21. SOLVING A MIXTURE PROBLEM Example 3 Charolette Besch is a chemist. She needs a 20% solution of alcohol. She has a 15% on hand as well as a 30% solution. How many liters of the 15% solution should she add to the 3L of 30% solution to obtain her 20% solution?

  22. SOLVING A MIXTURE PROBLEM Example 3 Solution Step 1 Read the problem. We must find the required number of liters of 15% alcohol solution.

  23. SOLVING A MIXTURE PROBLEM Example 3 Step 2 Assign a variable. Let x = the number of liters of 15% solution to be added. Sum must equal

  24. SOLVING A MIXTURE PROBLEM Example 3 Step 3 Write an equation. Since the number of liters of pure alcohol in the 15% solution plus the number of liters in the 30% solution must equal the number of liters in the final 20% solution. Liters in 15% Liters in 20% Liters in 30% + =

  25. SOLVING A MIXTURE PROBLEM Example 3 Step 4 Solve the equation. Liters in 15% Liters in 20% Liters in 30% + = Distributive property Subtract .60; subtract .15x. Divide by .05.

  26. SOLVING A MIXTURE PROBLEM Example 3 Step 5 State the answer. Thus, 6 L of 15% solution should be mixed with 3 L of 30% solution, giving 6 + 3 = 9 L of 20% solution. Step 6 Check. The answer checks since and

  27. Mixture Problems Problem Solving In mixed investment problems, multiply each principal by the interest rate to find the amount of interest earned.

  28. SOLVING AN INVESTMENT PROBLEM Example 4 An artist sold a painting for $410,000 and needs some of the money in 6 months and the rest in 1 year. Treasury bonds are available for 6 months at 4.65% and one for a year at 4.91%. A broker suggests that the two investments that will earn $14,961. How much should be invested at each rate to obtain that amount of interest?

  29. SOLVING AN INVESTMENT PROBLEM Example 4 Solution Step 1 Read the problem. We must find the amount to be invested at each rate.

  30. SOLVING AN INVESTMENT PROBLEM Example 4 Step 2 Assign a variable. Let x = dollar amount invested for 6 months at 4.65%; 410,000 – x = dollar amount invested for 1 year at 4.91%

  31. SOLVING AN INVESTMENT PROBLEM Example 4 Step 3 Write an equation. Interest from 4.65% investment Investment from 4.91% investment = Total interest +

  32. SOLVING AN INVESTMENT PROBLEM Example 4 Step 4 Solve the equation. Interest from 4.65% investment Investment from 4.91% investment = Total interest +

  33. SOLVING AN INVESTMENT PROBLEM Example 4 Step 5 State the answer. The artist should invest $200,000 at 4.65% for 6 months and $410,000 – $200,000 = $210,000 at 4.91% for 1 year to earn $14,961 in interest.

  34. SOLVING AN INVESTMENT PROBLEM Example 4 Step 6 Check. The 6-month investment earns… while the 1-yr investment earns The total amount of interest is

  35. Modeling with Linear Equations A mathematical model is an equation (or inequality) that describes the relationship between two quantities. A linear model is a linear equation.

  36. MODELING THE PREVENTION OF INDOOR POLLUTANTS Example 5 One of the most effective ways of removing contaminants such as carbon dioxide from the air while cooking is to use a vented range hood. If a range hood removes contaminants at a rate of F liters of air per second, then the percent P of contaminants that are also removed from the surrounding air can be modeled by the linear equation where

  37. MODELING THE PREVENTION OF INDOOR POLLUTANTS Example 5 What flow F must a range hood have to remove 50% of the contaminants from the air? Solution: Since P = 50, the equation becomes Subtract 7.18. Divide by 1.06. To remove 50% of the contaminants, the flow rate must be approximately 40.40 L of air per second.

  38. MODELING HEALTH CARE COSTS Example 6 The projected per capita health care expenditures in the United States, where y is in dollars, x in years after 2000, are given by the linear equation • What were the per capita health care expenditures in the year 2008? • When will the per capita expenditures reach $9000?

  39. MODELING HEALTH CARE COSTS Example 6 • What were the per capita health care expenditures in the year 2008? Let x = 8 and find the value of y. In 2008, the projected per capita health care expenditures were $7830.

  40. MODELING HEALTH CARE COSTS Example 6 (b) When will the per capita expenditures reach $9000? Let y = 9000 and find the value of x. 11 corresponds to 2000 + 11. Per capita health care expenditures are projected to reach $9000 by 2011.

More Related