b. Rewrite the formula isolating the variable P. In other words, write a formula for determining the value of P. P

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1 Math Applications The applications that follow are like the ones you will encounter in many workplaces. Use the mathematics you have learned in this chapter to solve the problems. Wherever possible, use your calculator to solve the problems that require numerical answers. 1 Social security tax is deducted from your paycheck using the formula T P G, where T is the tax amount to be deducted, P is the decimal percentage rate for the tax, and G is your gross pay. a. Write a sentence that describes how to compute the tax deducted from your paycheck. Multiply the rate and gross pay. b. Rewrite the formula isolating the variable P. In other words, write a formula for determining the value of P. P c. Suppose your paycheck stub shows gross pay of $ and a tax deduction of $ Determine the percentage rate used for computing your deduction. Round the answer to two decimal places. 7.51% 2 When a lever is used to lift or balance weights, as occurs in common patient scales seen in medical offices, the sizes and positions of the weights are governed by the equation F 1 D 1 D 2 Fulcrum F 2 F 1 D 1 F 2 D 2, where F 1 and F 2 are the weights (forces), and D 1 and D 2 are the distances from the fulcrum. a. You have a scale with a movable weight (F 2 ) that is 500 grams. The patient stands on a platform that is 0.2 cm (D 1 ) from the fulcrum. Write a formula you could use to compute the patient s weight (F 1 ) if the 500-gram weight is balanced at D 2 centimeters from the fulcrum. 500D 2 0.2F 1 or 2,500D 2 F 1 b. What is the weight of the person on the scale if the scale balances when you place the 500-gram weight 18 centimeters from the fulcrum? 45 kilograms Math Applications 181

2 3 Su is a technician for a drink dispensing company. She services drink machines in office buildings and cafeterias to ensure that they are functioning properly and dispensing the proper amount of drink. If a machine dispenses too much, it may overfill the customer s cup, and if the machine dispenses too little, customers may complain that they are not receiving the full amount. The drink dispensers offer a large 24-oz drink and a small 16-oz drink. a. The amount of drink dispensed should be within 2.5% of the amount advertised. What is 2.5% of the amount in a large drink? 0.6 oz b. What is 2.5% of the amount of drink in a small drink? 0.4 oz c. Let L represent the amount being dispensed for a large drink. Write and solve an absolute value equation for the minimum and maximum acceptable dispensed amounts. L ; L 23.4 oz, 24.6 oz d. Let S represent the amount being dispensed for a small drink. Write and solve an absolute value equation for the minimum and maximum acceptable dispensed amounts. S ; S 15.6 oz, 16.4 oz e. Su tests a machine by measuring the amounts dispensed for 5 large drinks and 5 small drinks. If all 10 amounts are within the minimum and maximum values specified, the machine does not need to be adjusted. Su s results are shown below. Large: 23.9 oz, 24.2 oz, 24.8 oz, 23.1 oz, 24.3 oz Small: 15.8 oz, 16.5 oz, 15.9 oz, 16.2 oz, 16.4 oz Does this particular machine need to be adjusted? yes 4 It costs a recording company $9,000 to prepare a CD for production. This represents a one-time fixed cost. Manufacturing, marketing, royalty payments, and so on are variable costs. Suppose the variable costs total $4.50 per CD. The CDs are sold in bulk to a record store for $6.75 each. How many CDs must be sold for the income from CD sales to equal the total costs? 182 Chapter 3 Solving Equations a. Write an equation for the income (I ), based on the number of CDs sold (N). I 6.75N b. Write an equation for the total costs (C), based on the number of CDs produced (N). C 9, N c. Set the expression for I equal to the expression for C, and then isolate the variable N to find out what value of N is needed to make I equal to C. Explain why your solution is reasonable. 4,000; the company makes just over $2 per CD sold; 4,000 $2 $8,000, which is close to $9,000.

3 5 You will receive a 5% commission on the sale of a condominium for a client. The client would like to obtain at least $120,000 from the sale. Use the formula below to determine the lowest selling price for the condominium that will allow the client to receive the amount asked for. (Round your answer to the nearest $100.) A (100% c) s A is the amount desired by the client from the sale. c is the commission that you will get from the sale. s is the selling price. $126,300 6 The Rule of 72 is a way to estimate the effect of compound interest on an investment. The rule states that you divide the annual percentage interest rate (expressed as a percent, not a decimal) into 72 to find the approximate number of years needed to double the value of your investment. a. Write the Rule of 72 as a formula. Be sure to identify the variables that you choose to use., where P is the rate as a percent and T is the time in years b. Use your formula to find out how long it would take to double a $500 deposit in an account that paid an annual interest rate of 8.5%. about 8.47 years c. Use your formula to find out what interest rate would be needed to double an investment in five years. 14.4% 7 Zach is going on vacation in Paris, France. He is getting ready to book a rental car for his arrival. The car rental company charges $75 plus $0.50 per mile. There is a 4% tax on the total charge. The car rental company also has a rate plan that charges $375 per week. a. Write a formula that can be used to compute the total cost of renting a car and driving m miles. Be sure to define your variables. How much does it cost to rent a car and drive 500 miles? C 1.04(75 0.5m) b. Rewrite the formula, isolating the total number of miles, m. c. Use the formula from part b to determine the number of miles Zach could drive to match the cost of renting a car for a week. How do you interpret the value for m? about 571; Driving 571 miles is about the same cost as the rate plan that charges $375 per week. Math Applications 183

4 8 You need to estimate the total cost for a job that requires a large volume of copying. After a phone call to the copy shop, you find that each copy on white paper costs $ On colored paper, it costs $ In addition, there is a $5 handling fee for jobs that exceed 500 copies. Each packet of materials that you will duplicate has 5 colored pages and 27 white pages. a. Write a formula that can be used to determine the total cost for producing n packets, when n is greater than 500. [27(0.045) 5(0.065)]n 5 c or 1.54n 5 c b. Suppose your supervisor tells you that this project has only $2,000 allotted for copying costs. Use your formula to determine the number of packets you can produce with this allotted amount. 1,295 9 You are told that the demand for a certain product can be described by the equation D 4,000 44C (for values of C between 10 and 90). D is the demand, in pounds, for the product. C is the cost per pound in cents. In addition, the supply of this product from the distributor is described by the equation S 1, C (for values of C between 10 and 90). S C is the supply, in pounds, of the product. is the cost per pound in cents. When the supply exceeds the demand, the price will go down. On the other hand, when the demand exceeds the supply, the price will go up. At what cost per pound will the price stabilize and the supply equal the demand? 30 cents per pound 10 As children grow, dosages for medications gradually approach those for adults. Clark s rule is often used to determine the correct dosage for children. Child s dose Adult dose Weight of child in pounds 150 pounds a. What dosage should be administered to a child weighing 28 pounds if the adult dose of a certain drug is 80 milligrams? Round your answer to the nearest whole milligram. 15 milligrams b. How much would a child have to weigh to receive the adult dosage? Explain why your answer is reasonable. 150 pounds; When a child weighs 150 pounds, the multiplying ratio is 1. A 150-pound child weighs about the same as a smallsized adult. 184 Chapter 3 Solving Equations

5 11 A person s vital capacity is the amount of air that can be exhaled after one deep breath. An estimate of the vital capacity for women can be obtained by the formula V 0.041h 0.018A V h A is the vital capacity in liters. is the woman s height in centimeters. is the woman s age in years. a. Determine an estimated vital capacity for yourself or a friend using the formula. Answer will vary. b. Suppose that by using a spirometer, the lung capacity of a healthy woman is measured at 3.4 liters. The woman is 168 centimeters tall. Isolate the variable A in the formula. Use the given data to find a value for A and complete the sentence that follows: This woman exhibited the breathing capacity of a? -year-old woman When a fair-skinned person comes into a tanning salon, the receptionist normally advise him or her to start with 15 minutes exposure, then gradually build up to a maximum of 23 minutes, increasing the exposure 1 minute every days. a. Which of the following equations would describe the exposure time, T (in minutes), for a fair-skinned client who has been tanning regularly for D days, prior to the point when he or she reaches the maximum time? iv i. T 23D 15 ii. T 2.5D 2 23 iii. T 15D 2.5 iv. T 0.4D 15 b. How much exposure does the equation suggest is recommended on the 5th day of treatment? On the 10th day? 17 minutes; 19 minutes c. Solve the equation for D and use this new equation to determine how many days a client needs to reach the maximum 23 minutes of exposure. see margin Math Applications 185

6 13 The following table lists permissible sound exposures in the workplace established by the Occupational Safety and Health Administration (OSHA). Hours of Exposure Weighted per Day Sound Level (decibels) A computer analysis of the data generates the equation S 2.817H S is the maximum permissible sound level in decibels. H is the number of hours the person is in this environment. a. Use the equation to compute the maximum possible sound levels for the hours of exposure listed in the table. Compare the computed sound levels to the values in the table. 86.4; 92; 97.6; 100.4; 103.3; 104.7; 106.1; 107.5; b. Do you think the equation provides a good description of the table? yes 14 You are the manager of a drug testing laboratory. You want to compare the cost of breeding your own mice rather than purchasing them from a supply company. You can buy 6-week-old mice from the supply company for $4.90 each. Alternately, you estimate the cost of breeding the mice would be $830 each week in overhead and $0.60 per mouse each week for food. You need to find the number of mice that would be needed each week to reach a point where the cost of purchasing equals the cost of breeding. a. How many mice would the lab need per week for the decision to breed the mice to be cost-effective? 193 b. If more mice than this number are needed per week, would it be cheaper to purchase the mice or to breed them? breed the mice 186 Chapter 3 Solving Equations

7 15 Cosmetologists are frequently paid by the salary and commission method. The cosmetologist is paid a base salary plus a percentage of any money received that exceeds twice this base salary. The situation is modeled by the following equation G S C(I 2S). G is the gross pay (in dollars) for the period if I is greater than twice the salary amount. S is the salary amount for the period. C is the commission percentage rate (a decimal percent). I is the gross income for services performed. a. Suppose your weekly salary is $475, with a 15% commission on all sales over twice your salary. Use the formula to determine your gross pay if the gross income for your services totaled $1,185 during this week. Explain why your solution is reasonable. $510.25; the income is somewhat greater than the weekly salary. b. How much income for services is needed to have a gross pay of $750? $2, You examine the cost of bus travel for your vacation. After analyzing the fare schedule, you make a table of costs and mileages. Mileage One-way Fare 35 $ $ $ $ 23 Estimate the cost of a one-way fare for travel to a city that is 50 miles away. about $ Suppose you consider two job offers. One has a starting salary of $32,900 with raises of $1,500 per year. The second offer is for a starting salary of $37,500 with raises of $950 per year. a. Will these two salaries ever be the same? If so, when? yes; after about 8.4 years b. You hope to qualify for a promotion and a significant increase in salary after six years. Which of the two starting jobs would give you the higher salary at the end of six years? second job Math Applications 187

8 18 You are designing an exhaust fan for a small garage by using an electric motor connected to a large fan blade by a fan belt and two pulleys. Fan pulley Belt Motor pulley The approximate length of belt needed to connect the two pulleys is given by the formula B 2L 2 (D d ). B is the length of the belt. L is the distance between the two pulley centers. D is the diameter of the larger pulley. d is the diameter of the smaller pulley. a. The formula with definitions for the variables does not specify any units. You should assume the equation is true as long as the units from the variables are the same. If L, D, and d are measured in meters, what is the unit for B? meters b. Compute the approximate length of belt needed to connect the two pulleys if the two pulley centers are 3 feet apart. The larger pulley is 8 inches in diameter, and the smaller pulley is 3.5 inches in diameter. about 90 inches 19 Ohm s Law states that the voltage (in volts) is equal to the current (in amperes) times the resistance (in ohms). a. Write Ohm s Law as an equation. Use V for the voltage, i for the current, and r for the resistance. V ir b. Suppose you observe various currents through a 3,300-ohm resistor. Rewrite your equation for this condition. V i 3,300 c. Use the equation from part b to determine what voltage you should measure if the current in the 3,300-ohm resistor is ampere volts 188 Chapter 3 Solving Equations

9 20 In close-up photography, the distance of an object from the lens determines how far the lens must be from the film. This requires special lenses and focusing mechanisms. These distances (all measured in the same units) are related by the formula 1 f 1 1 q. p f p q is the focal length of the lens. is the distance of the object being viewed from the center of the lens. is the distance of the image formed on the film from the center of the lens. q p f image focal point lens object You have a 50-millimeter close-up lens. With this lens, the maximum distance you can position the lens from the film plate is 6.2 centimeters. You need to find the closest you can position an object to the lens to get a sharp photograph. a. Rewrite the formula, isolating the variable representing the object distance. see margin b. Determine the object distance predicted by the equation for a lens with a focal length of 50 millimeters and a maximum image distance of 6.2 centimeters centimeters 21 You need to make a wire-wound resistor with a resistance of 15 ohms using 22-gauge copper wire. From a table, you find that 22-gauge copper wire has a resistance of ohm per centimeter length. Thus, you can determine the resistance of a wire by multiplying the length of the wire by the number of ohms per centimeter. a. Determine the resistance of 75 feet of 22-gauge copper wire. about 30.9 ohms b. Rewrite your formula, isolating the variable for length. Use this new formula to determine what length of 22-gauge copper wire you need to create the 15-ohm wire-wound resistor. L 5 ; about 1, centimeters Math Applications 189

10 22 A particular electronics circuit lists several resistances R 2, R 3, and R 4 that are based on R 1. You cannot measure R 1 but you can measure R 2 it measures 110 ohms. Use this measurement to determine what the resistance for R 1 must be. Also determine what R 3 and R 4 must be. Resistor Resistance (ohms) R R R R 1 R R 1 23 The National Fire Codes have guidelines for storing flammable liquids. An amount of 2 gallons per square foot is permitted in a storeroom with no fire protection (sprinkler or other type) and a rated fire resistance of 1 hour. The storeroom cannot be larger than 150 square feet. a. Write an equation to express the relationship between the gallons of flammable liquid that can be stored and the area of the storeroom. G 2F b. Identify the variables and the constants in your equation. G and F are variables; 2 is a constant c. Use the equation to calculate the floor area required to store a maximum of 165 gallons square feet d. How many gallons can you store in the largest storeroom? You need to calibrate a variable, wire-wound resistor. You find that when the contact is at the minimum position, your resistor has a resistance of 4 ohms. Then, for every centimeter of travel away from this minimum position, the resistance increases by 2.5 ohms. This relationship continues up to a maximum of 29 ohms. At this resistance, the contact is at the farthest position 10 centimeters from the minimum. a. Write a formula to compute the resistance of the resistor based on the distance from the minimum position. R 4 2.5d b. How can you be sure of the validity of your formula? (10) c. Isolate the variable for the distance of the contact. Where should you place the contact to obtain a resistance of 22 ohms? 7.2 centimeters 190 Chapter 3 Solving Equations

11 25 You are writing a contract for commercial sandblasting before repainting a building. To estimate the cost of the sandblasting, you determine the sand cost and the labor cost (to operate the sandblaster). You find that the sand costs $4 per bag and that it takes about 10 minutes to load and use each bag of sand. The labor cost for the operator is $40 per hour plus a one-time $110 equipment fee. Find an equation to estimate the total cost of the sandblasting, depending on the time. a. Write an equation for the cost of the equipment used by an operator who works T hours. c T b. Determine how much sand would be used in an hour (60 minutes) and hence how much it would cost per hour. Write an equation for the cost of sand when sandblasting for T hours. C 24T; 6 bags; $24 per hour c. Add the cost of the labor and equipment to the cost of the sand to find the total cost of sandblasting for T hours. C T d. Use your equation to estimate the total cost if the job takes 2 hours. What is the cost for a 4-hour job and for a 6-hour job? $238; $366; $ You work in a metal shop and must often drill equally spaced holes along a length of stock. The holes on the ends of the stock must be centered exactly 1 inch from the ends. The remaining holes will be equally spaced. An example is shown below. 1" S S 1" L a. Write a formula for the spacings of the holes on a piece of stock that is L inches long and that must have a total of N equally spaced holes. see margin b. What spacing between holes (their centers) would be needed for 9 holes on a bar that is 10 feet long? inches c. Suppose you use a 3 -inch 4 drill bit (i.e., which has a 3 -inch 4 diameter). Describe the layout equatio of the holes on the bar if your equation yielded a spacing value of 0.75 inches. For nine holes in the bar, what length of stock would yield this value for the spacing? 8 inches Math Applications 191