Applied Mathematics. Level 7. Worldwide Interactive Network, Inc Waterford Place, Kingston, TN

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1 Applied Mathematics Level 7 Worldwide Interactive Network, Inc Waterford Place, Kingston, TN Worldwide Interactive Network, Inc. All rights reserved.

2 Copyright 1998 by Worldwide Interactive Network, Inc. ALL RIGHTS RESERVED. Printed in the U.S.A. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, photocopying, recording or otherwise without the prior written permission of Worldwide Interactive Network, Inc. ACT and WorkKeys are trademarks of ACT, Inc. Worldwide Interactive Network, Inc. is not owned or controlled by ACT, Inc.; however, ACT, Inc. has reviewed these training materials and has determined that they meet ACT, Inc. s standards for WorkKeys Training curriculum. The WorkKeys employment system is a product of ACT, Inc. The use of materials in this manual does not imply any specific results when WIN materials are used with the ACT WorkKeys system. Requests for permission to reproduce or make other use of this material should be addressed to: Worldwide Interactive Network, Inc Waterford Place Kingston, Tennessee USA Tel: (865) Fax: (865) info@w-win.com 2 Applied Mathematics

3 INTRODUCTION Hi, I m EdWIN Hi, my name is EdWIN. I will be your guide through Applied Mathematics Level 7. Together we will proceed through this course at your speed. Look for me to pop up throughout your lessons to give you helpful tips, suggestions, and maybe even a pop quiz question or two. Don t worry, you can find the answers to them at the end of the course. If the content of the lesson is something you understand, you should be able to work through it at a faster pace. On the other hand, if the material is difficult, read the text several times and then try to work the exercises one at a time. After you try one problem, look at the solution. You can learn by reviewing each step that is provided in the solution and by concentrating on the process being illustrated. Now let s think positive; no negative attitudes allowed!! Applied Mathematics 3

4 INTRODUCTION Level 7 of Applied Mathematics is designed to help you develop problem solving skills that can be applied in various situations. Following a good problem solving strategy will be stressed in this level. Most of the problems require analyzing the given information to determine the necessary steps involved in solving the problem. Many of the problems require four or more steps. Interpretation of graphic representations is needed to solve some problems. The concept of percent should be a familiar idea to you at this level, but this level extends the concept of percent to include problems dealing with increases and decreases. Finding volume of three-dimensional figures such as cylinders, spheres, and cones and working with irregular shapes are required in some problems. You will also be required to make cost effective decisions based on comparison of two calculations which may involve percentage differences, unit cost, or the interpretation of graphical information. When you complete this level you should be able to find and correct mistakes made in a solution to a problem from any of the previous levels. This is the last level. You ve done well to get this far. Good luck! 4 Applied Mathematics

5 OUTLINE LESSON 1 LESSON 2 LESSON 3 LESSON 4 LESSON 5 LESSON 6 LESSON 7 LESSON 8 LESSON 9 REFERENCES Review of Prerequisite Skills Solving Problems Involving Percent of Change Proportions Perimeter, Area, and Volume Solving Multistep Problems Using Graphs and Charts to Solve Problems Multistep/Decision Problems Finding Mistakes Posttest Workplace Problem Solving Glossary Test-Taking Tips Formula Sheet Applied Mathematics 5

6 LESSON 1 REVIEW OF PREREQUISITE SKILLS Before we begin Level 7, we should refresh our skills from previous levels. The following pretest contains problems which emphasize the skills you should know. The answers are provided following the pretest. You must determine if you are ready to proceed with Level 7 by how well you solve the problems. Good Luck! Ready, set, go! 6 Applied Mathematics

7 LESSON 1 EXERCISE PRETEST Instructions: Perform the indicated operations using your calculator as needed. Use 3.14 to approximate π = = = = = = (-12) = = = = = = = = Applied Mathematics 7

8 LESSON What is the area of a rectangle 7 in by 8 in? 16. What is the perimeter of a rectangle 7 in by 8 in? 17. What is the area of a circle with a diameter of 15 cm? 8 Applied Mathematics

9 LESSON What is the volume of a rectangular solid 7 m by 8 m by 4 m? is what percent of 50? 20. Convert 88% to a decimal and a fraction. Applied Mathematics 9

10 LESSON 1 Instructions: Read the following problems. Determine if there are errors in the solving process or calculations. 21. You have a piece of pipe 1 meter long. You need to cut two pieces that are each 37 centimeters long. After cutting the pieces, how many centimeters of pipe is left?(assume no waste during cutting.) Given answer: 63 cm of pipe left 22. A machine can produce 576 units in 18 hours. How many units will it produce in 28 hours? Given answer: 875 units 10 Applied Mathematics

11 LESSON An assembly line moves at a rate of 50 objects/min. If the rate goes up by 4 objects/min for 6 minutes, then slows down by 5 objects/ min for 9 minutes before resuming normal speed, how many objects have passed a worker stationed on the line the last hour? Given answer: 3, A car dealership is advertising a used Toyota Tercel for only $400 down and $170 per month for 4 years. What is the total cost of the car? Given Answer: The total cost of the car would be $8, Joe Black receives a salary of $2,240 per month. If 18% is deducted for income tax and 6% is deducted for savings, how much does Joe take home? Given answer: $1, Applied Mathematics 11

12 LESSON 1 ANSWERS TO EXERCISE = = Answer: 103 Answer: = = Answer: -16 Answer: = = Answer: Answer: (-12) = = Answer: 19 Answer: = = Answer: 9 8 or Answer: = = 2 Answer: 5 24 Answer: 6 5 or Applied Mathematics

13 LESSON = = Answer: 65 6 or Answer: or What is the area of a rectangle 7 in by 8 in? Answer: 56 in What is the perimeter of a rectangle 7 in by 8 in? Answer: 30 in 17. What is the area of a circle with a diameter of 15 cm? Answer: π (7.5) 2 = cm What is the volume of a rectangular solid 7 m by 8 m by 4 m? Answer: 224 m is what percent of 50? Answer: 17 = N% = N% 34% 20. Convert 88% to a decimal and a fraction. Answer: 88% = = = Applied Mathematics 13

14 LESSON You have a piece of pipe 1 meter long. You need to cut two pieces that are each 37 centimeters long. After cutting the pieces, how many centimeters of pipe is left?(assume no waste during cutting.) Given answer: 63 cm of pipe left To solve the problem, first convert 1 meter to centimeters. 1 m = 100 cm Key words how much left (which indicates subtraction) Mistake = 63 Supposedly, there are 63 cm of pipe left. Answer: The metric conversion was correct. The assumption that subtraction is needed to solve the problem was correct. An error was made when only one piece of pipe 37 cm was subtracted from 100. The problem indicated 2 pieces of pipe 37 cm were cut, so 37 2 = 74 cm of pipe needs to be subtracted from = 26 cm of pipe left is the correct answer. 14 Applied Mathematics

15 LESSON A machine can produce 576 units in 18 hours. How many units will it produce in 28 hours? Given answer: 875 units To solve the problem, use a proportion = 28 x Find x. 576x = x = 504 x =.875 Supposedly, 875 units are produced in 28 hours. Answer: A proportion is used to solve the problem, but it was set up incorrectly. If units over hours are used in the first ratio (fraction), then the hours given (28) must be placed on the bottom of the second ratio. The correct answer follows: = x 28 16,128 = 18x 896 = x 896 units are produced in 28 hours is the correct answer. Applied Mathematics 15

16 LESSON An assembly line moves at a rate of 50 objects/min. If the rate goes up by 4 objects/min for 6 minutes, then slows down by 5 objects/ min for 9 minutes before resuming normal speed, how many objects have passed a worker stationed on the line the last hour? Given answer: 3,015 To solve the problem, draw a diagram to step through the process. This allows you to see what mathematical operations need to be calculated. 2, = 3,015 objects in one hour Answer: This is the correct answer. 24. A car dealership is advertising a used Toyota Tercel for only $400 down and $170 per month for 4 years. What is the total cost of the car? Given Answer: The total cost of the car would be $8,016. To solve the problem, convert years to months. 4 yr 12 mo = 48 months So, your payments will be $170 for 48 months. $ = $8,016 The cost of the car is $8,016. Answer: This process was correct, but the calculation $ was incorrect = 8,160. The solution also neglected to include the original $400 down in the final cost. So, $400 + $8,160 = $8,560 is the correct answer. 16 Applied Mathematics

17 LESSON Joe Black receives a salary of $2,240 per month. If 18% is deducted for income tax and 6% is deducted for savings, how much does Joe take home? Given answer: $1, To solve the problem, calculate 18% of 2,240 and 6% of 2,240: 18% 2240 = $ % 2240 = $ Total deductions are $ $ = $ This amount must be subtracted from Joe s salary. $2,240 - $ = $1, Alternate method to solve the problem: Calculate the total percentage being deducted: 18% + 6% = 24% 24% (amount deducted) of $2,240 (total salary) = $ This amount must be subtracted from Joe s salary. $2,240 - $ = $1, take home Answer: This is the correct answer. Applied Mathematics 17

18 LESSON 1 This page was intentionally left blank 18 Applied Mathematics

19 LESSON 2 SOLVING PROBLEMS INVOLVING PERCENT OF CHANGE Let s begin with a concept that has been discussed briefly in a previous level. This concept is called percent of change. This means you will learn to determine how much something increases or decreases in a situation. For example, if you are buying clothing that was $35.00 last week and $25.00 this week, you might need to know the percentage that the price increases. To do this, we use a formula. Percent Increase or Decrease = the amount of increase / decrease original amount In our example, the price fell by $ This is the amount of the decrease. The clothing originally cost $ This is the original amount. $ $ =. 285 = 29% The price of clothing decreased by 29%. Applied Mathematics 19

20 LESSON 2 Sometimes you may have to eliminate some information to do a particular problem. For example, take a look at this problem. As a contractor was looking over a new house plan, he decided that the 15 ft by 18 ft den needed to be enlarged by 4 ft in each direction. He also considered enlarging the 12 ft by 8 ft kitchen. The finished house had a total square footage of 3,485 square feet, excluding the garage. By what percent did he increase the area of the den? When you look at this problem, we see it only asks us about the area of the den. The den s area was originally = 270 sq ft. The new area, when the den is enlarged by 4 ft in each direction, is = 418 sq ft. Using our formula for change after we find the amount of increase, 418 (enlarged square footage) (original square footage) = 148 (amount of increase) 148 amount of increase 270 original square footage = 55% rounded tonearest whole number This means that the den s size increased by 55%. We did not need to know the square footage of the house nor the kitchen. This shows that you may have to sort through the information before you do the problem. Now, try to do some problems of this type on your own. You may need to look back at the formula and the examples. Good luck! 20 Applied Mathematics

21 LESSON 2 Pop Quiz: You want to plant 18 tulip bulbs in your front yard. You may purchase individual bulbs, but they are priced 5 for $4.90. How much will you spend to buy the bulbs for your yard? Applied Mathematics 21

22 LESSON 2 EXERCISE PERCENT OF CHANGE Instructions: Solve the following percentage problems. Round final answers to the nearest whole number. 1. You borrow $5,860 from a bank. You will have monthly payments of $ for 3 years. What percent will you pay back in interest over the entire period of the loan? 2. In January, during the 1st quarter of the year, Miller & Company shipped 22,000 units of its product. In February, the company shipped 35,000 units of its product. In March, 8,000 employees went on strike for a month after shipping 3,000 units. In the 4th quarter of the previous year, the company shipped 75,000 units. What is the percent of change from the 4th quarter of last year to the 1st quarter of this year? Is this an increase or a decrease? 22 Applied Mathematics

23 LESSON 2 3. Cantrell s, a jewelry store on 34th Street, is having a watch sale. Watches that originally sold for $ each are on sale for $ The store has 10 of these watches on sale. What is the percent of price change? 4. A home in Wedgewood Estates, a 20-home subdivision, has an asking price of $165,000. The present owner paid $100,000 for the home and spent $15,000 in improvements. The current buyer agrees to pay $152,000. What is the percent of increase based on the seller s investment in the home and the buyer s price? Applied Mathematics 23

24 LESSON 2 5. The Chattanooga Choo Choo provides services for 5,000 tourists in a 30 day (month) period. The next month they provide services for 6,400 tourists in a 31 day (month) period. What is the percent of increase in tourists from one month to the next? Catch that train! 24 Applied Mathematics

25 LESSON 2 ANSWERS TO EXERCISE 1. You borrow $5,860 from a bank. You will have monthly payments of $ for 3 years. What percent will you pay back in interest over the entire period of the loan? Answer: 26% 3 years = 36 months $ = $7, (you pay over 3 years) $7, $5,860 = $1, $1,523.60(amount increase) $5,860 (original amount) =.26 = 26% 2. In January, during the 1st quarter of the year, Miller & Company shipped 22,000 units of its product. In February, the company shipped 35,000 units of its product. In March, 8,000 employees went on strike for a month after shipping 3,000 units. In the 4th quarter of the previous year, the company shipped 75,000 units. What is the percent of change from the 4th quarter of last year to the 1st quarter of this year? Is this an increase or a decrease? Answer: 20% decrease 22, , ,000 = 60,000 units in 1st quarter 75,000 units in 4th quarter 60,000-75,000 = -15,000 The negative sign indicates a decrease. 15,000(amount decreased) 75,000(original amount) =.20 = 20% decrease Applied Mathematics 25

26 LESSON 2 3. Cantrell s, a jewelry store on 34th Street, is having a watch sale. Watches that originally sold for $ each are on sale for $ The store has 10 of these watches on sale. What is the percent of price change? Answer: 35% price reduction $ $ = $ or $ $950 = -$ (decrease) =.35 = 35% price change 4. A home in Wedgewood Estates, a 20-home subdivision, has an asking price of $165,000. The present owner paid $100,000 for the home and spent $15,000 in improvements. The current buyer agrees to pay $152,000. What is the percent of increase based on the seller s investment in the home and the buyer s price? Answer: 32% increase Seller s investment price : $100,000 + $115,000 = $115,000 Buyer s price: $152,000 $152,000 - $115,000 = $37,000 difference $37,000 $115,000 =.32 = 32% increase 26 Applied Mathematics

27 LESSON 2 5. The Chattanooga Choo Choo provides services for 5,000 tourists in a 30 day (month) period. The next month they provide services for 6400 tourists in a 31 day (month) period. What is the percent of increase in tourists from one month to the next? Answer: 28% increase in tourists 6,400-5,000 = 1,400 more in second month 1,400 5,000 =.28 = 28% increase in tourists Applied Mathematics 27

28 LESSON 3 PROPORTIONS Lesson 3 begins with a review of proportions. Remember that a proportion is defined as two equal ratios. For example, 3 4 = 9 is a proportion since = 9 4. You can solve a proportion when part of it is unknown by using cross multiplication. For example: 3 4 = 9 N N 3N = 9 4 3N = 36 3N = N = 12 We have seen these problems in previous levels of Applied Mathematics, so you should be familiar with these problems already. Solve the following four problems to refresh your memory. Different letters may be used to represent the unknowns. 28 Applied Mathematics

29 LESSON 3 EXERCISE PROPORTION REVIEW Instructions: Use cross multiplication to solve the proportions. 1. x 4 = x = = 4. x 7 = N Applied Mathematics 29

30 LESSON 3 ANSWERS TO EXERCISE 1. x 4 = x 4 = 3 Answer: 9x = 60 Answer: 4x = 24 x = 6.7 x = = 4. x 7 = N Answer: 50x = 400 Answer: 90N = 1260 x = 8 N = Applied Mathematics

31 LESSON 3 We can use these proportions to solve practical problems in the workplace. For example, you might need to use proportions to calculate the gas you will need on a trip. Let s suppose you know that to travel 50 miles in the company car, you need to buy 3 gallons of gas. You know that your next trip will be around 120 miles. You also know that your boss expects an estimate of how much gas you will need. You should set up a proportion. Keep in mind that you should be consistent. If you put the number of miles on the top on one side of the equal sign, miles should be on top for the other ratio. Which way do I go? 50N = N 50 = N = 7.2 gal You will need to purchase 7.2 gallons of gas. Now you try the next few problems on your own. I will write the answers on the following page. Applied Mathematics 31

32 LESSON 3 EXERCISE REVIEW OF PROPORTION PROBLEMS Instructions: Solve the following problems using cross multiplication. 1. Sheryl, the office manager, can type 89 words per minute. How long would it take her to type a document containing 1,045 words? 2. A trip 50 miles out of town takes 45 minutes. Driving at the same rate, how long would it take to drive another 120 miles? 32 Applied Mathematics

33 LESSON 3 3. Fertilizer must be mixed with water in a 1:4 ratio. If you use 3 cups of fertilizer, how much water would be needed? 4. If it takes 8 gallons of semi-gloss paint to paint three offices, how many gallons will it take to paint 21 offices? Applied Mathematics 33

34 LESSON 3 ANSWERS TO EXERCISE 1. Sheryl, the office manager, can type 89 words per minute. How long would it take her to type a document containing 1,045 words? Answer: 89words 1 045words =, 1min x min 89 x = 1,045 x = 11.7 minutes 2. A trip 50 miles out of town takes 45 minutes. Driving at the same rate, how long would it take to drive another 120 miles? Answer: 50 miles 120 miles = 45 min x min 50 x = 5,400 x = 108 minutes 3. Fertilizer must be mixed with water in a 1:4 ratio. If you use 3 cups of fertilizer, how much water would be needed? Answer: 1 part fertilizer 3 cups fertilizer = 4 partswater x cupswater x = 12 cups of water 4. If it takes 8 gallons of semi-gloss paint to paint three offices, how many gallons will it take to paint 21 offices? Answer: 8 gal x gal = 3 offices 21 offices 3x = 168 x = 56 gallons 34 Applied Mathematics

35 LESSON 3 Sometimes you may have to deal with more than one set of (given) information... For example: A contractor recommends installing 2 air vents for every 350 square feet of floor space on the first floor, but he recommends 3 air vents for every 400 square feet of floor space on the second floor. You have a 3,000 square foot house, and 40% of the floor space is on the second floor. How many vents do you need on each level? Let s look at this problem; we see a rate for the first floor ( 2 vents 3 vents ) and a rate for the second floor ( 350 sq ft 400 sq ft ). In this problem, you must also calculate a percentage: 40% of 3, ,000 sq ft = 1,200 sq ft. This is the square footage for the second floor. Now, let s set up a proportion. Second Floor 3vents 400sq ft x vents = 1,200sq ft 400x = 3600 x = 9 vents recommended for the second floor Applied Mathematics 35

36 LESSON 3 If you had 1,200 sq ft on the second floor and 3,000 sq ft total, there must be 3,000-1,200 = 1,800 sq ft on the first floor. Now let s calculate the number of vents needed for the first floor. First Floor 2 = x x = 3600 x = 10.2 = 11 vents on the first floor Sometimes you have to use all of the information to get an answer. Think through each problem very carefully to decide what information you will need. Why don t you try some proportion applications on your own? 36 Applied Mathematics

37 LESSON 3 EXERCISE PROPORTION PROBLEMS Instructions: Solve the following problems using our strategies to solve word problems and cross multiplication. Round final answers to the nearest hundredth. 1. The scale on a blueprint indicates 1 inch represents 6 feet. A wall mural costs $15.95 per linear foot. A builder has been asked to estimate the cost to apply the mural to a wall that measures inches on the blueprint. What will be the cost estimate? 2. A pharmacist working at a hospital received a request for 1 dose of medication for a patient who weighs 185 lb. The dosage for the medication requested is 5 ml per 25 kg of body weight. The cost of the medication to the pharmacist is $1.30 per milliliter. If there is a 15% markup on medication, how much did the pharmacist charge the patient s account? (1 lb = 0.45 kg) Applied Mathematics 37

38 LESSON 3 3. In your fertilizer business, you must place an order for nitrogen. You have already determined that the proper ratio for your mixture is 24% nitrogen, 4% phosphate, 8% potassium, and 64% bulk mixer. You will combine these components in a mixer with a 5 ton capacity and you want to order enough nitrogen to mix 10 batches. The nitrogen sells for $200 per ton. If the order is for more than 1 full ton, there is a $15 discount on each full ton. If the order is for more than 2 full tons, there is an additional $5 discount per each ton. How much will you pay for this order of nitrogen? 4. A group of students from the Student Educator Club and other students from the Fa La La Chorus held a charity carnival 3 months ago. They took in $890 at the carnival, but they used $340 for expenses. The Student Educator Club had 8 students who each worked a 5 hour shift at the carnival, and the chorus had 4 people who each worked 7 hours. If the profits are divided up proportionally according to how many hours students from each group worked at the carnival, how much would each group receive? 38 Applied Mathematics

39 LESSON 3 ANSWERS TO EXERCISE 1. The scale on a blueprint indicates 1 inch represents 6 feet. A wall mural costs $15.95 per linear foot. A builder has been asked to estimate the cost to apply the mural to a wall that measures inches on the blueprint. What will be the cost estimate? Answer: Calculate the length of the wall: (change fraction to decimal) 1 inch 6 feet inches = x feet 1x = x = feet Determine the cost: $ $ Applied Mathematics 39

40 LESSON 3 2. A pharmacist working at a hospital received a request for 1 dose of medication for a patient who weighs 185 lb. The dosage for the medication requested is 5 ml per 25 kg of body weight. The cost of the medication to the pharmacist is $1.30 per milliliter. If there is a 15% markup on medication, how much did the pharmacist charge the patient s account? (1 lb = 0.45 kg) Answer: Convert the patient s weight to kg: 1lb 185 lb = 0.45 kg xkg 1x = So, the patient weighs kg. Determine the correct dosage for this patient. How many ml does the patient need? 5ml xml = 25 kg kg 25x = x = ml The correct dosage is ml. Calculate the cost: $ = $21.65 pharmacist s cost 15% $21.65 = $3.25 markup $ $3.25 = $24.90 The patient was charged $24.90 for the medication. 40 Applied Mathematics

41 LESSON 3 3. In your fertilizer business, you must place an order for nitrogen. You have already determined that the proper ratio for your mixture is 24% nitrogen, 4% phosphate, 8% potassium, and 64% bulk mixer. You will combine these components in a mixer with a 5 ton capacity and you want to order enough nitrogen to mix 10 batches. The nitrogen sells for $200 per ton. If the order is for more than 1 full ton, there is a $15 discount on each full ton. If the order is for more than 2 full tons, there is an additional $5 discount per each ton. How much will you pay for this order of nitrogen? Answer: Several steps are needed to solve this problem. Calculate total of fertilizer needed: 10 (batches) 5 (ton per batch) = 50 tons Calculate percent of nitrogen: 24% of 50 (tons) = 12 tons There are 12 tons of nitrogen in 50 tons of fertilizer. Calculate cost: (before discount) 12 (tons) $200 (per ton) = $2,400 Calculate discount: more than 1 ton: $15 (per ton) 12 (tons) = $180 discount more than 2 tons: $5 (per ton) 12 (tons) = $60 additional discount together: $180 + $60 = $240 Cost with discount: $2,400 - $240 = $2,160 You should pay $2,160 for the nitrogen. Applied Mathematics 41

42 LESSON 3 4. A group of students from the Student Educator Club and other students from the Fa La La Chorus held a charity carnival 3 months ago. They took in $890 at the carnival, but they used $340 for expenses. The Student Educator Club had 8 students who each worked a 5 hour shift at the carnival, and the chorus had 4 people who each worked 7 hours. If the profits are divided up proportionally according to how many hours students from each group worked at the carnival, how much would each group receive? Answer: Several steps are needed to solve this problem. Calculate total hours worked: Club 8 5 = 40 hours Chorus 4 7 = 28 hours = 68 total hours Calculate profit: $890 - $340 (expenses) = $ 550 (profit) Club Share: 40 club hours 68 total hours 68x = 22,000 x = $ x = $ Applied Mathematics

43 LESSON 3 Chorus Share: 28 chorus hours 68 total hours x = $550 68x = x = $ I think I will help work the carnival next time! Applied Mathematics 43

44 LESSON 4 Moving on to geometric figures... PERIMETER, AREA, AND VOLUME Previous levels of Applied Mathematics introduced perimeter, area, and volume. These measures are frequently used in the workplace and are therefore, objectives assessed by the ACT WorkKeys Applied Mathematics assessment. Let s review the formulas for some common geometric figures before we review calculations of perimeter, area, and volume. 2 Here are some review problems. I ll leave a space for you to work, and then you can peek at my work to check your answers. After that, I ll leave you some word problems to practice application of perimeter, area, and volume. 44 Applied Mathematics

45 LESSON 4 EXERCISE PERIMETER, AREA, AND VOLUME REVIEW Instructions: Use the appropriate formulas to calculate each problem. Round answers to the nearest tenth. Use the approximation 3.14 for π. 1. Find the volume of a rectangular solid 5 cm 4 cm 3 cm. 2. Find the circumference of a circle with a radius of 4 in. 3. Find the area of a circle with a diameter of 6 m. 4. Find the perimeter of a rectangle with a length of 14 cm and a width of 12 cm. 5. Find the area of a triangle with a base of 12 ft and a height of 10 ft. Applied Mathematics 45

46 LESSON 4 ANSWERS TO EXERCISE Note: Multiplication is primarily indicated be the symbol in this course. However, parentheses are sometimes used to imply multiplication, particularly when calculating formulas. 1. Find the volume of a rectangular solid 5 cm 4 cm 3 cm. Answer: V = l w h V = V = 60 cm 3 or 60 cubic centimeters 2. Find the circumference of a circle with a radius of 4 in. Answer: C = πd C = π(8) radius is half of diameter C = 25.1 in 3. Find the area of a circle with a diameter of 6 m. Answer: A = πr 2 A = π(3) 2 A = 28.3 m 2 4. Find the perimeter of a rectangle with a length of 14 cm and a width of 12 cm. Answer: P = 2( ) P = 52 cm 5. Find the area of a triangle with a base of 12 ft and a height of 10 ft. Answer: A= 1 2 b h A = 1 2 (12) (10) A = 60 ft 2 Now, let s take a look at word problems. 46 Applied Mathematics

47 LESSON 4 EXERCISE APPLICATION OF PERIMETER, AREA, AND VOLUME Instructions: Solve the following word problems. Round your final answers to the nearest hundredth. Use the approximation of 3.14 for π. 1. The kitchen and dining areas of the Backman house are adjacent and together they form a 14 ft by 16 ft rectangle. The Backman family wants to install wood flooring in three-fifths of this area. What is the area to be refloored? 2. One 50 pound package of grass seed is enough to seed 10,000 square feet of ground. Your lawn is 400 feet by 285 feet. How many packages of grass seed are needed? Applied Mathematics 47

48 LESSON 4 3. You need to buy edging to border a round flower bed which is three feet in diameter. How much edging is needed? 4. Concrete is to be poured into a frame that is 12 ft by 6 ft by 6 in. (Concrete is poured by the cubic yard.) How much concrete is needed? 5. What is the volume of a box which measures 4 ft by 5 ft by 8 in? 48 Applied Mathematics

49 LESSON 4 ANSWERS TO EXERCISE 1. The kitchen and dining areas of the Backman house are adjacent and together they form a 14 ft by 16 ft rectangle. The Backman family wants to install wood flooring in three-fifths of this area. What is the area to be refloored? Answer: = sq ft or sq ft 2. One 50 pound package of grass seed is enough to seed 10,000 square feet of ground. Your lawn is 400 feet by 285 feet. How many packages of grass seed are needed? Answer: = 114, ,000 10,000 = packages of grass seed 3. You need to buy edging to border a round flower bed which is three feet in diameter. How much edging is needed? Answer: C = πd C = π(3) C = 9.42 ft of edging Remember that circumference and perimeter are outside edges and area is inside surface. Applied Mathematics 49

50 LESSON 4 4. Concrete is to be poured into a frame that is 12 ft by 6 ft by 6 in. (Concrete is poured by the cubic yard.) How much concrete is needed? Answer: First, change all measurements to yards. 12 ft 1yd 3ft 6 ft 1yd 3ft 6 in 1 yd 36 in = 4 yd = 2 yd =.17 yd Calculate volume V = l w h V = V = 1.36 cubic yards Note Because we rounded the decimal when we converted 6 in to yd (.17), we have created a margin of error. If we had rounded to thousandths instead of hundredths, our error would have been smaller,..., to tenthousandths even smaller, etc. We can avoid this error if we convert to fractions instead of decimals. (Measures in fractions can be precise since you do not round fractions.) But, you may not be as comfortable working in fractions as decimals. 50 Applied Mathematics

51 LESSON 4 Problem 4 worked using fractions (to avoid rounding error) follows: First change all measurements to yards. 12 ft 1yd 3 ft = 4 yd 6 ft 1yd 3 ft = 2 yd 6 in 1yd 36 in = 6 36 = 1 6 yd Calculate volume: V = l w h V = V = cubic yards = Notice a slight difference in this answer and 1.36 calculated using decimals = which rounds to 1.33 cubic yards and 1.36 cubic yards are close, but not the same answer. The decision to round is dependent upon the accuracy required by the situation you are trying to solve. (continued) Applied Mathematics 51

52 LESSON 4 This course is primarily designed to assist you in your mathematics problem solving ability, not focusing on improving basic math skills. Use your calculator to perform math operations with decimals and fractions. Since calculators vary in the rounding process, exercise assignments will request you round to certain decimal places in some problems. This is to avoid our different calculators indicating slightly different answers. Remember, rounding creates error. I will use decimals in solving many problems in this course since everyone s calculator handles decimals. Please be aware, rounding causes error. 5. What is the volume of a box which measures 4 ft by 5 ft by 8 in? Answer: Convert 8 inches to feet: 8 in 1ft =.67 ft ( ) 12in = cubic feet This problem could also be calculated in inches. 12 in 4 ft 1ft = 48 in 12 in 5 ft = 60 in 1ft = 23,040 cubic inches 52 Applied Mathematics

53 LESSON 4 We ve briefly reviewed volume of rectangles. Now, let s look at the volume of some other shapes. The formulas are given. Let s look at a cone, a cylinder, and a sphere. We will approximate π as 3.14 and round numbers to the nearest hundredth in our final answers. Let s look at some simple problems first. Geometric shapes are in our workplaces. Applied Mathematics 53

54 LESSON 4 Find the volume of a cone with a radius of 2 in and a height of 4 in. V = πrh 2 3 V = π(2) 2 ( 4) 3 V = V = V = in 3 Note When you check your answers with mine, do not be alarmed if they are not exact. Since we are likely using calculators which round to different levels of precision, our answers may vary. On most calculators, if you use the π key, the answer to this problem is However, if you do not have a calculator with the π key, you may use 3.14, which is an approximation for π. If you use 3.14, the answer to this problem is Because we are approxmating π, we are creating a margin of error. In order to reduce the margin of error as much as possible, avoid rounding your final answer. While we are discussing approximation through rounding, please be aware that the directions for the ACT WorkKeys Applied Mathematics assessment instructs examinees to use the approximation of 3.14 for π. 54 Applied Mathematics

55 LESSON 4 Find the volume of a cylinder with a radius of 6 cm and a height of 4 cm. V = πr 2 h V = π (6) 2 (4) V = V = cm 3 Find the volume of a sphere with a radius of 12 ft V = 4 π 3 r 3 V = 4 π( 3 12 ) 3 V = ,728 V = 7, ft 3 Pop Quiz: Your production line reports 14 defective parts on Monday, 7 defective parts on Tuesday, 8 parts rejected on Wednesday, 5 on Thursday, and 11 defective parts on Friday. On the average, how many defective parts does your line produce per day? Applied Mathematics 55

56 LESSON 4 Now, let s use these formulas in a word problem: A wine vat in the shape of a right circular cylinder has a base diameter of 14 in and a height of 2 ft. How many cubic feet of wine will the tank hold? How many gallons of wine will it hold? (1 cubic foot 7.5 gallons) In this problem, you must use formulas and make some conversions that you learned previously. Cylinder Base is the bottom diameter (14 in) and half of this (7 in) will be the radius. First, convert 7 in to ft: 7 in 1 ft/12 in = 7/12 ft Calculate volume: V = πr 2 h V = 3.14 ( 7 12 ) 2 2 V = 2.14 ft 3 Cheers! or V = πr 2 h V = 3.14 (.58) 2 2 V = 2.11 ft 3 56 Applied Mathematics

57 LESSON 4 Notice the slight difference due to rounding decimals. This will cause variance in the second part of the problem. The first answer (using fractions) will be closer to the true volume, so I will use it in the second part of our problem. This answers the first part of our problem, how many cubic feet will the vat hold. Now, how many gallons of wine will it hold? 2.14 ft gal = gal 3 1ft Now, take a look at the next four problems. Some may seem difficult, but you can do it. Do not give up too quickly. You will have at least one multistep volume problem on the ACT WorkKeys Applied Mathematics assessment. Applied Mathematics 57

58 LESSON 4 EXERCISE VOLUME PROBLEMS Instructions: Solve the following word problems. Round your final answers to the nearest hundredth. Use the approximation of 3.14 for π. 1. Water is placed in a cylindrical bucket on a bench. The bucket has a diameter of 12 in and a height of 12 in. The bench is capable of supporting a maximum of 200 lb. Water weighs about 64.2 lb per cubic foot. Will the bench support the bucket when it is full? 2. You are pouring canned sodas into a cylindrical pitcher. Each can of soda is 12 cm tall and has a diameter of 6.5 cm. The pitcher is 36 cm tall and has a diameter of 20 cm. How many cans of soda will the pitcher hold? 58 Applied Mathematics

59 LESSON 4 3. Jeff s hot rod show car has a spherical gas tank. The diameter of the tank is 24 in. About how many gallons of fuel will the tank hold? 4. Water for a small town is stored in a spherical tank with a diameter of 30 ft. The city uses 1 full tank per day. If the city must make $1,500 per tank to cover its costs, what must it charge per 100 gallons of water? Applied Mathematics 59

60 LESSON 4 ANSWERS TO EXERCISE 1. Water is placed in a cylindrical bucket on a bench. The bucket has a diameter of 12 in and a height of 12 in. The bench is capable of supporting a maximum of 200 lb. Water weighs about 64.2 lb per cubic foot. Will the bench support the bucket when it is full? Answer: Find volume of cylindrical bucket V = πr 2 h Convert all measures to ft: r = 6 in 1ft 12 in h = 12 in = 1 foot =.5 ft Calculate volume: V = πr 2 h V = 3.14 (.5) 2 1 V =.79 ft 3 or.79 cubic feet Calculate weight of water: Given 64.2 lb per cubic foot ft 3 = lb Will the bench hold it? Is it smaller than 200 lb? Yes, the bench will hold it. 60 Applied Mathematics

61 LESSON 4 2. You are pouring canned sodas into a cylindrical pitcher. Each can of soda is 12 cm tall and has a diameter of 6.5 cm. The pitcher is 36 cm tall and has a diameter of 20 cm. How many cans of soda will the pitcher hold? Answer: Cylindrical V = π (r) 2 h cans: V = π (3.25) 2 12 = 398 cm 3 pitcher: V = π (10) 2 36 = 11,304 cm 3 pitcher cans 11, or 28 cans 3. Jeff s hot rod show car has a spherical gas tank. The diameter of the tank is 24 in. About how many gallons of fuel will the tank hold? Answer: Calculate radius: Diameter is 24 in which makes the radius ( 1 2 of the diameter) 12 in Convert to feet: 12 in 1 ft 12 in = 1ft Calculate volume of the sphere: V = 4 3 πr3 V = (1)3 V = ft 3 Applied Mathematics 61

62 LESSON 4 Calculate gallons of fuel: Remember 1 cubic foot 7.5 gallons (continued) ft gal 1ft 3 = 31.4 gallons 4. Water for a small town is stored in a spherical tank with a diameter of 30 ft. The city uses 1 full tank per day. If the city must make $1,500 per tank to cover its costs, what must it charge per 100 gallons of water? Answer: Calculate the volume of the sphere: V = 4 3 πr3 V = 4 3 π (15) 3 V = 14,130 ft 3 Find how many gallons the sphere will hold: 1ft gal 14,130 ft 3 7.5gal 1ft 3 105,975 gallons Find charge per 100 gallons of water: $1,500 gal 105,975 gal $x = 100 gal 105,975x = 150, Applied Mathematics

63 LESSON 4 x = $1.42 per 100 gallons of water Sometimes we have to use more than one perimeter and area formula to solve a particular problem. A shape may consist of a combination of rectangles, circles, and/ or triangles. Example: Ron plans to hire a lawn service to mow his yard. It costs $0.06 per square meter to have the lawn mowed. Using the diagram of Ron s lot, calculate the cost of the mowing service. This is a multistep problem in which we will calculate several areas. Notice I first divided the lot with dashes to form several rectangles and a triangle. Applied Mathematics 63

64 LESSON 4 I will first find the area of the rectangle labeled A. Area A : = 494 m 2 Now, calculate the area of the house and subtract it from Area A. Area of the house: = 221 m 2 Area A of the yard that is shaded: = 273 m 2 We now must calculate the area at the other end of the lot. If we subtract 24.7 from 38.2, we see that Area B is 13.5 m by 13.6 m. Area B : = m 2 To find the area of the triangle, Area C, we must do some calculations. If we extend the backside of the lot to make a rectangle, we can subtract 13.6 from 20 (same length as the front side) to find the base of the triangle = 6.4 m The height of the triangle should be 13.5 m since it is a side of rectangle B. The area of a triangle is 1 2 bh. A = 1 2 bh Area C = = 43.2 m2 64 Applied Mathematics

65 LESSON 4 Now, back to our problem. We need to add these areas to determine how many square meters need to be mowed. (I could have already mowed the yard by the time I figure out how much this is going to cost!) = m 2 total area to be mowed The cost is $0.06 per square meter. Cost to have Ron s yard mowed: = $29.99 Ron... taking it easy while his yard is being mowed. The following three problems have shapes that you will have to modify to do the problems. Try these problems. You can always peek at my work if you need help. Applied Mathematics 65

66 LESSON 4 EXERCISE AREA AND PERIMETER PROBLEMS Instruction: Solve the following problems. You may need to refer to some formulas for area or perimeter of various shapes. 1. What is the perimeter of a symmetrical star with 5 points if each side of the star is 18 inches in length? Speaking of stars... I compete in rodeos when I am not teaching. 66 Applied Mathematics

67 LESSON 4 2. A track is made in the shape shown. How much fencing will be needed to build a fence around the outside of the track? Applied Mathematics 67

68 LESSON 4 3. Aluminum siding costs $17.50 per square yard. How much will it cost to buy enough siding for the front and back sides of the barn in the diagram? (Assume no door in the back) 68 Applied Mathematics

69 LESSON 4 ANSWERS TO EXERCISE 1. What is the perimeter of a symmetrical star with 5 points if each side of the star is 18 inches in length? Answer: 180 inches 18 inches 10 sides = 180 in 2. A track is made in the shape shown. How much fencing will be needed to build a fence around the outside of the track? Answer: 81.4 feet Divide the oval into a rectangle 25 by 10 and 2 semicircles. The semicircles at the ends of the track together form a circle with a diameter of 10 ft. Circle: C = πd C = π(10) = 31.4 ft 25 (top side) + 25 (bottom side) (circumference) = 81.4 ft Applied Mathematics 69

70 LESSON 4 3. Aluminum siding costs $17.50 per square yard. How much will it cost to buy enough siding for the front and back sides of the barn in the diagram? (Assume no door in the back) Answer: $651 STEP 1 Gable area A = 1 2 bh A = A = 37.5 ft (sides) = 75 ft 2 STEP 2 Front and back area A = l w A = A = 150 ft (front and back) = 300 ft 2 70 Applied Mathematics

71 LESSON 4 STEP 3 Door area A = l w A = 8 5 A = 40 ft = 335 ft 2 1 square yard = 9 square feet 1yd ft 2 2 = yd 9ft = $ cost of siding Pop Quiz: The company cafeteria charges $3.95 for a sandwich lunch which includes a sandwich, chips, and medium drink. Next week they are providing a 5-day sandwich ticket for $9.25 (sandwiches only). If chips are 50 and a medium drink is 75, what is the least amount you could pay for a sandwich, chips, and medium drink for 5 days? Applied Mathematics 71

72 LESSON 5 SOLVING MULTISTEP PROBLEMS Sometimes you must work several steps to a problem with different situations for each step. For example, you might be paying someone $5.50 an hour for the first 3 hours of work and $6.00 for every hour after that. If the person works 5 hours, you will have to pay $ $ = $ $12.00 = $28.50 This problem required calculations to change as the situation changed. Example: If you buy one gross (144) of tape dispensers, it costs $576. If you buy more than one gross, you receive a 5% discount for every gross after the first one, up to a maximum of 25%. For example, 2nd gross = 5% discount / 3rd gross = an additional 5% discount (overall discount thus far =10%), etc. If 4 gross of dispensers are ordered, what is the unit cost of each dispenser? The price changes depending on how many dispensers you order. In this problem you order 4 gross of the tape dispensers. This means you pay $576 for the first gross. You should receive a discount on the additional 3 gross purchased. 72 Applied Mathematics

73 LESSON 5 GROSS CUMULATIVE DISCOUNT 1 gross none or more 5% 10% 15% 20% 25% To calculate the discount: $ X 3 = $1,728 original cost $1,728 X 15% = $ $1,728 - $ (discount) = $1, cost of 3 discounted gross Calculate total: $ (1st gross) + $1, (3 additional gross) = $2, (cost of 4 gross of tape dispensers) Calculate unit price of dispensers: Number of units bought = 576 Cost divided by the number of units bought $2, (cost) / 576 (units bought) = $3.55 per unit Each tape dispenser costs $3.55 for this order. For extra practice you might try calculating orders for a different amount of dispensers. Applied Mathematics 73

74 LESSON 5 EXERCISE MULTISTEP PROBLEMS Instructions: Solve the following problems. 1. A shirt was originally marked $35, but it is on sale this week for 25% off. Because you are an employee at the department store, you receive an extra 10% discount. What is the resulting discount you receive? Is this the same as a 35% discount? Did someone say, Sale? 74 Applied Mathematics

75 LESSON 5 2. There were 20 people in an in-service class at the office. At the end of the class, the instructor gave a 50 question test to determine if each person should receive in-service credit. The scores and frequency of scores are given in the table. What was the average score? Applied Mathematics 75

76 LESSON 5 3. You operate a machine that bends 10 mm pieces of wire into paper clips. Occasionally, the machine produces a reject that will be recycled. The number of rejects varies with the production rates which are indicated in the figure below. If the machine is set to produce 400 paper clips per hour, approximately how many paper clips do you have to produce to meet your quota of 1,600 good 76 Applied Mathematics

77 LESSON 5 ones? ANSWERS TO EXERCISE 1. A shirt was originally marked $35, but it is on sale this week for 25% off. Because you are an employee at the department store, you receive an extra 10% discount. What is the resulting discount you receive? Is this the same as a 35% discount? Answer: Employee Discount $35(original price) 25% = $8.75 $35(original price) - $8.75(25% off) = $26.25 Additional 10% is off the sale price $ % = $2.63 $26.25(sale price) - $2.63(10% off) = $23.62 employee discount price Actual 35% off sale $35(original price) 35% = $12.25 $35(original price) - $12.25(35% off) = $22.75 Applied Mathematics 77

78 LESSON 5 The 35% discount is not the same discount as the employee receives. 2. There were 20 people in an in-service class at the office. At the end of the class, the instructor gave a 50 question test to determine if each person should receive in-service credit. The scores and frequency of scores are given in the table. What was the average score? Answer: 1, = 88.7 average score 78 Applied Mathematics

79 LESSON 5 3. You operate a machine that bends 10 mm pieces of wire into paper clips. Occasionally, the machine produces a reject that will be recycled. The number of rejects varies with the production rates which are indicated in the figure below. If the machine is set to produce 400 paper clips per hour, approximately how many paper lips do you have to produce to meet your quota of 1,600 good ones? c Answer: This problem first requires an exact understanding of what the question is asking: how many paper clips, in addition to the 1,600 that you need to make your quota, must be produced in order to cover the number of rejects at that production rate? Examine the graph to see that rejects are based per 100 paper clips. At 400 per hour, approximately 9 rejects would occur. To determine how many good paper clips we will produce at this rate, calculate: 100 (total) - 9 (rejects) = 91 (good paper clips produced) Applied Mathematics 79

80 LESSON 5 (continued) We can now use a proportion to find out how many paper clips we must produce to reach our quota of 1,600 good ones. 91(good) 100(total) 1,600(good) = x (total) 91x = 160,000 x = 1, You must produce 1,759 paper clips to have 1,600 good ones. 80 Applied Mathematics

81 LESSON 6 USING GRAPHS AND CHARTS TO SOLVE PROBLEMS In the last exercise, you were required to read a line graph to work the problem. It is often necessary to read graphs and charts to solve problems in the workplace. So, let s talk about this more. When you read a graph, you look for specific information you need to solve the problem. Let s begin with a circle graph which is also called a pie graph. Let s say, for example you are interested in food. (I always am!) You see that 25% of this budget goes for food. If I told you that your total budget for a month was $1,500, you could determine how much would be set aside for food. $1,500 25% = $375 See how simple that is! I ve chosen two problems for you to do involving charts or graphs. Peek at my Applied Mathematics 81

82 LESSON 6 work only if you get stuck. EXERCISE CHARTS AND GRAPHS I nstructions: Use the following graphics to solve these problems. Round answers to the nearest whole number. 1. A lady left $750,000 to 5 of her grandchildren and greatgrandchildren. The percentage each received is shown on the graph. Compare the amount of Joyce s inheritance to Mickey s. 82 Applied Mathematics

83 LESSON 6 2. There are 60 people in a Teamwork and Cooperation class offered on weekends at work. The class is divided into 4 groups. Each group s average test score is given in the chart. What was the Applied Mathematics 83