Bridge-Thickness Experiment. Student 2

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1 Applications 1. Below are some results from the bridge-thickness eperiment. Bridge-Thickness Eperiment Thickness (laers) Breaking Weight (pennies) a. Plot the (thickness, breaking weight) data. Draw a line that models the pattern in the data. b. Find an equation for the line ou drew. c. Use our equation to predict the breaking weights of paper bridges, 5, and 7 laers thick.. Which line do ou think is a better model for the data? Eplain. Student 1 Student. Cop each graph onto grid paper. Draw a line that fits each set of data as closel as possible. Describe the strategies ou used. Graph A Graph B Graph C Investigation Linear Models and Equations

2 . This table gives the average weights of purebred Chihuahuas from birth to 1 weeks. a. Graph the (age, weight) data. Draw a line that models the data pattern. b. Write an equation of the form = m + b for our line. Eplain what the values of m and b tell ou about this situation. c. Use our equation to predict the average weight of Chihuahuas for odd-numbered ages from 1 to 15 weeks. d. What average weight does our linear model predict for a Chihuahua that is 1 weeks old? Eplain wh this prediction is unlikel to be accurate. 5. U-Wash-It Car Wash did market research to determine how much to charge for a car wash. The compan makes this table based on its findings. U-Wash-It Projections For: Help with Eercise 5 Web Code: ape-15 Price per Wash $ $5 $1 $15 $ Customers Epected per Da a. Graph the (price, epected customers) data. Draw a line that models the data pattern. b. Write an equation in the form = m + b for our graph. Eplain what the values of m and b tell ou about this situation. c. Use our equation to estimate the number of customers epected for prices of $.5, $7.5, and $1.5. Thinking With Mathematical Models

3 . Find the slope, -intercept, and equation for each line. a. 1 b. 1 (, 1) (, ) 1 (, ) 1 (, ) 1 1 c. 1 d. (, 9) (, 9) 1 (, 1) (, ) 1 1 The relationships in Eercises 7 1 are linear. 7. a. A tpical American bab weighs about pounds at birth and gains about 1.5 pounds per month for the first ear of life. What equation relates weight w in pounds to age a in months? b. Can this model be used to predict weight at age? Eplain.. Kaa bus a $ phone card. She is charged $.15 per minute for long-distance calls. What equation gives the value v left on her card after she makes t minutes of long-distance calls? 9. Dakota lives 1,5 meters from school. She leaves for school, walking at a speed of meters per minute. Write an equation for her distance d in meters from school after she walks for t minutes. 1. A car can average 1 miles on 5 gallons of gasoline. Write an equation for the distance d in miles the car can travel on g gallons of gas. Investigation Linear Models and Equations 5

4 11. Write a linear equation for each table relating and. a. b c. d For Eercises 1 17, find an equation for the line that satisfies the conditions. 1. Slope.; -intercept (,.) 1. Slope -intercept (, 5) ; 1. Slope ; passes through (, 1) 15. Passes through (, 15) and (5, ) 1. Passes through (-, ) and (5, -) 17. Parallel to the line with equation = 15 - and passes through (, ) For: Multiple-Choice Skills Practice Web Code: apa Write an equation for each line. 1 O 1 Thinking With Mathematical Models

5 19. Anchee and Jonah earn weekl allowances for doing chores over the summer. Anchee s father pas her $5 each week. Jonah s mother paid him $ at the beginning of the summer and now pas him $ each week. The relationships between number of weeks worked and dollars earned are shown in this graph. Earnings From Chores $5 $ Earnings $ $ $1 $ Number of Weeks 1 a. Which line represents Jonah s earnings? Which line represents Anchee s earnings? Eplain. b. Write two linear equations in the form = m + b to show the relationships between Anchee s earnings and the number of weeks she works and between Jonah s earnings and the number of weeks he works. c. What do the values of m and b in each equation tell about the relationship between the number of weeks and the dollars earned? d. What do the values of m and b tell about each line? Investigation Linear Models and Equations 7

6 For Eercises, do the following: a. Solve the equation. Show our steps. b. Graph the associated line (for eample, for 5.5 ± 57, graph 5.5 ± ). Label the point that shows the solution = = =. 7 = 5-7. At Water Works Amusement Park, the dail profit from the concession stands depends on the number of park visitors. The equation p =.5v - 5 gives the estimated profit p in dollars if v people visit the park. In parts (a) (c), use a graph to estimate the answer. Then, find the answer b writing and solving an equation or inequalit. a. For what number of visitors will the profit be about $,? b. One da people visit the park. What is the approimate concession-stand profit for that da? c. For what number of visitors will the profit be at least $5? 5. The following formulas give the fare f in dollars that two bus companies charge for trips of d miles. Transcontinental: f =.15d + 1 Intercit Epress: f = 5 +.d In parts (a) (c), use a graph to estimate the answer. Then, find the answer b writing and solving an equation or inequalit. a. For Transcontinental, how man miles is a trip that costs $99? b. For Intercit Epress, how far can a person travel for a fare that is at most $99? c. Is there a distance for which the fare for the two bus lines is the same? If so, give the distance and the fare. Solve each equation. Show the steps in our solutions = = = Find at least three values of for which the inequalit is true. 9. # 1., # # 1 Thinking With Mathematical Models

7 . Ever Frida, the mechanic for Columbus Public Schools records the miles driven and the gallons of gas used for each school bus. One week, the mechanic records these data. Data for Columbus Bus Fleet Bus Number Gas Used (gal) Miles Driven a. Write a linear equation that models the relationship between miles driven d and gallons of gas used g. b. Use our equation to predict the number of miles such a bus could travel on 1 gallons of gas. c. Use our equation to predict the number of gallons of gas required to drive such a bus 5 miles. d. What do the values of m and b in our equation d = mg + b tell about the fuel efficienc of the school bus fleet?. One of the most popular items at a farmers market is sweet corn. This table shows relationships among the price for the corn, the demand for the corn (how much corn people want to bu), and the leftovers of corn (how much corn the market has at the end of the da). Sweet Corn Suppl and Demand Price per Dozen $1 $1.5 $. $.5 $. $.5 Demand (dozens) Leftovers (dozens) a. Wh do ou think the demand for corn decreases as the price goes up? b. Wh do ou think the leftovers of corn increases as the price goes up? c. Write a linear equation that models the relationship between demand d and price p. d. Write a linear equation that models the relationship between leftovers / and price p. e. Use graphs to estimate the price for which the leftovers equals the demand. Then, find the price b solving smbolicall. Investigation Linear Models and Equations 9

8 Connections 5. Tell whether each table represents a linear relationship. Eplain. a b c For parts (a) (d), cop the table. Then, use the equation to complete the table. Tell whether the relationship is linear. Eplain. a. =- - b. = ( - 7) c. = ( + ) d. = - 1 Cop each pair of numbers in Eercises 7. Insert <, >, or to make a true statement Thinking With Mathematical Models

9 . Madeline sets a cop machine to enlarge b a factor of 15%. She then uses the machine to cop a polgon. Write an equation that relates the perimeter of the polgon after the enlargement a to the perimeter before the enlargement b. For Eercises 5, evaluate the epression without using a calculator (-7) (-15) (-5) 5. (-) (-.5) 5. -? (-.5) 5. You can epress the slope of a line in different was. The slope of the line below is, or.. You can also sa the slope is % because the 1 rise is % of the run. 1 These numbers represent slopes of lines. % % % a. Which numbers represent the same slope? b. Which number represents the greatest slope? Which represents the least slope? Investigation Linear Models and Equations 1

10 5. Consider the following stories and the graphs. a. Match each stor with a graph. Tell how ou would label the aes. Eplain how each part of the stor is represented in the graph. Stor 1 A parachutist is taken up in a plane. After he jumps, the wind blows him off course. He ends up tangled in the branches of a tree. Stor Ella puts some mone in the bank. She leaves it there to earn interest for several ears. Then one da, she withdraws half of the mone in the account. Stor Gerr has a big pile of gravel to spread on his drivewa. On the first da, he moves half of the gravel from the pile to his drivewa. The net da he is tired and moves onl half of what is left. The third da he again moves half of what is left in the pile. He continues in this wa until the pile has almost disappeared. Graph A Graph B Graph C Graph D b. One of the graphs does not match a stor. Make up our own stor for that graph. Thinking With Mathematical Models

11 55. The figures below are similar..1 in.. in. A 1. in. 1. in. B 1 in. a. Find. b. What is the scale factor from Triangle A to Triangle B? c. What is the scale factor from Triangle B to Triangle A? d. How are the scale factors in parts (b) and (c) related? Etensions 5. A bridge-painting compan uses the formula C = 5, + 15L to estimate painting costs. C is the cost in dollars, and L is the length of the bridge in feet. To make a profit, the compan increases a cost estimate b % to arrive at a bid price. For eample, if the cost estimate is $1,, the bid price will be $1,. a. Find bid prices for bridges 1 feet, feet, and feet long. b. Write a formula relating the final bid price to bridge length. c. Use our formula to find bid prices for bridges 15 feet, feet, and 5 feet long. d. How would our formula change if the markup for profit was 15% instead of %? 57. Recall that Custom Steel Products builds beams from steel rods. Here is a 7-foot beam. 7-foot beam made from 7 rods a. Which of these formulas represents the relationship between beam length O and number of rods r? r = O r = O + (O - 1) + O r = (O - 1) + r = O - 1 b. How might ou have reasoned to come up with each formula? Investigation Linear Models and Equations

12 5. Recall that Custom Steel Products uses steel rods to make staircase frames. Here are staircase frames with 1,, and steps. 1 step made from rods steps made from 1 rods steps made from 1 rods Which of these formulas represents the relationship between the number of steps n and number of rods r? r = n + n r = n(n + ) r = n + r = (n + )n Custom Steel Products builds cubes out of square steel plates measuring 1 foot on a side. At right is a 1-foot cube. Use this information for Eercises How man square plates are needed to make a 1-foot cube?. Multiple Choice Suppose CSP wants to triple the dimensions of the cube. How man times the number of plates in the original cube will the need for this larger cube? A. B. C. D Multiple Choice Suppose CSP triples the dimensions of the original cube. How man times the volume of the original cube is the volume of the new cube? F. G. 9 H. 7 J. 1 1 ft 1 ft 1 ft. At Yvonne s Auto Detailing, car washes cost $5 for an time up to 1 minutes, plus $. per minute after that. The managers at Yvonne s are tring to agree on a formula for calculating the cost c for a t-minute car wash. a. Sid thinks c =.t + 5 is correct. Is he right? b. Tina proposes the formula c =.(t - 1) + 5. Is she right? c. Jamal told Tina her formula could be simplified to c =.t + 1. Is he right? Thinking With Mathematical Models

13 . Write an equation for each relationship. a. One tai compan charges $1.5 for the first miles of an trip, and then $1. for each mile after that. How is the tai fare related to the distance of a trip? b. An airport offers free parking for minutes and then charges $. for each hour after that. How is the price for parking related to the time a car is parked? c. A local cinema makes $.5 on each ticket sold. However, it has operating epenses of $75 per da. How is dail profit related to number of tickets sold? d. Rush Computer Repair sends technicians to businesses to fi computers. The charge a fied fee of $5, plus $5 per hour. How is total cost for a repair related to time the repair takes? Investigation Linear Models and Equations 5