Friedrich Von Steuben Metropolitan Science Center



Name__________________________________ Class__________________________ Date_______________________2-3FOCUS ON MODELINGModeling with One-Variable Linear Equations and InequalitiesCOMMON CORECC.9-12.N.Q.1*,CC.9-12.N.Q.2*,CC.9-12.A.CED.1*,CC.9-12.A.CED.2*,CC.9-12.A.CED.3*,CC.9-12.F.BF.1*Essential question: How can you use linear equations and inequalities to analyze a weight-loss plan? Jed is trying to devise a plan to help him lose weight. The weight-loss plan will be supervised by his doctor. His doctor explained to Jed that each pound of body fat is equal to 3500 Calories. So, if Jed eliminates 500 Calories per day through diet and exercise, he will lose one pound per week. The table below shows how many Calories are burned per hour of exercise for people of different weights who are walking or running at various speeds.Calories Burned per Hour of Exercise by Body WeightBody WeightExercise (1 hour)130 lb155 lb180 lb205 lbWalking 2.0 mph148176204233Walking 3.0 mph195232270307Walking 4.0 mph295352409465Running, 5.0 mph472563654745Running, 6.0 mph590704817931Running, 6.7 mph6497748991024Running, 7.5 mph73888010221163Running, 8.6 mph82698511441303Running, 10.0 mph944112613081489Jed currently weighs 205 pounds. Using a pace calculator, he has determined that he walks at a speed of 3 miles per hour and runs at a speed of 5 miles per hour. How can Jed use the table to devise a plan for losing weight? He plans to exercise 30 minutes per day.??1???Write a linear function for the total number of Calories burned in each session.?A? Since Jed is exercising 30 minutes per day, divide each value in the table by 60 to determine the number of Calories burned per minute. Round to the nearest hundredth.Calories Burned per Minute of Exercise by Body WeightBody WeightExercise (1 min.)130 lb155 lb180 lb205 lbWalking, 2.0 mph???2.47??????2.93??????3.40??????3.88???Walking, 3.0 mph???3.25??????3.87??????4.50??????5.12???Walking, 4.0 mph???4.92??????5.87??????6.82??????7.75???Running, 5 mph???7.87??????9.38?????10.90????12.42??Running, 6 mph???9.83?????11.73????13.62????15.52??Running, 6.7 mph??10.82????12.90????14.98????17.07??Running, 7.5 mph??12.30????14.67????17.03????19.38??Running, 8.6 mph??13.77????16.42????19.07????21.72??Running, 10 mph??15.73????18.77????21.80????24.82???B? Let t be the number of minutes Jed walks at 3.0 miles per hour. Write an expression for the number of Calories Jed burns while walking.5.12t???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????C??If Jed exercises for 30 minutes, then 30 – t is the number of minutes Jed runs at 5.0 miles per hour. Write an expression for the number of Calories Jed burns while running.12.42(30 – t)??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????D? The number of Calories burned while Jed exercises is given below in words. Use this verbal model and your expressions from parts B and C above to write a function C(t), where C is the total number of Calories burned.Total number ofCalories burned=Calories burnedwhile walking+Calories burnedwhile running???????????????????C(t)????????????????????=????????????????5.12t??????????????????+????????????12.42(30 ? t)?????????????????????????????????????????????????????????????????????????????????E? State the domain of the function.{t | 0 ≤ t ≤ 30}??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????REFLECT??1a. Why did you divide the rates in the original table by 60 to determine the rates in the second table?To convert the values in the original table from Calories per hour to??????????????????????????????????????????????????????????????????????????????????????????????Calories per minute, multiply by the conversion factor ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????1b. What unit of measurement does each term of C(t), 5.12t and 12.42(30 – t), have? How do you know?Calories; the coefficients 5.12 and 12.42 are measured in Calories per minute, ?????????????????????????????????????????????????????????????????????????????while the variables t and 30 – t are measured in minutes, so the products 5.12t ??????????????????????????????????????????????????????????????????????????????and 12.42(30 – t) are measured in = Calories.??????????????????????????????????????????????????????????????????????????????????????????????????????1c. What unit of measurement do the values of C(t) have? How do you know?Calories; C(t) is the sum of two terms measured in Calories, and???????????????????????????????????????????????????????????????????????????????????????????????????????Calories + Calories = Calories.????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ??2???Determine whether it is possible for Jed to burn 500 Calories per day from exercise alone.?A? Solve the equation C(t) = 500 for t. Justify the steps.????5.12t + 12.42(30 – t) = 500??????????Given equation?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????5.12t + 372.6 ? 12.42t = 500??????????Distributive Property ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????7.3t + 372.6 = 500??????????Distributive Property (combining like terms)??????????????????????????????????????????????????????????????????????????????????????????????????????7.3t = 127.4????????Subtraction Property of Equality??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????t ≈ ?17.5????????Division Property of Equality??????????????????????????????????????????????????????????????????????????????????????????????????B? Interpret the solution. Is it possible for Jed to burn 500 Calories per day by exercising for 30 minutes? Explain.Since t represents time spent walking, t must be nonnegative; the value of t in ??????????????????????????????????????????????????????????????????????????????????part A is negative, so it is not possible to burn 500 Calories in 30 minutes.????????????????????????????????????????????????????????????????????????????????????????????REFLECT??2a. The possible values of t range from 0 to 30. For what value of t does Jed burn the fewest number of Calories? For what value of t does Jed burn the greatest number of Calories? Explain your reasoning.t = 30 results in the fewest number of Calories burned because Jed spends all?????????????????????????????????????????????????????????????????????????????????????of his exercise time walking, which burns fewer Calories than running does;????????????????????????????????????????????????????????????????????????????????????????t = 0 results in the greatest number of Calories burned because Jed spends all????????????????????????????????????????????????????????????????????????????????????of his exercise time running.??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????2b. Find the value of C(0). How is this number relevant to the question of whether Jed can burn 500 Calories per day by exercising for 30 minutes?C(0) = 372.6; this represents the greatest number of Calories that Jed can??????????????????????????????????????????????????????????????????????????????????????????burn, and it is less than his goal of 500 Calories.????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ??3???Revise Jed's exercise plan so that he burns 500 Calories per day.?A? If he wants to burn 500 Calories per day, one option that Jed has is to exercise for more than 30 minutes. Suppose he decides to exercise for 45 minutes. Give the new rule for C(t).C(t) = 5.12t + 12.42(45 ? t)????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????B? Solve the equation C(t) = 500 for t. Show and justify the steps that you take.????5.12t + 12.42(45 ? t) = 500????????Substitute the rule for C(t).????????????????????????????????????????????????????????????????????????????????????????????????????????5.12t + 558.9 ? 12.42t = 500?????????Distributive Property???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????7.3t + 558.9 = 500?????????Distributive Property (combining like terms) ??????????????????????????????????????????????????????????????????????????????????????????????????????7.3t = ?58.9??????Subtraction Property of Equality??????????????????? ?????????????????????????????????????????????????????????????????????????????????????????????????????????????t ≈ 8?????????????Division Property of Equality?????????????????????????????????????????????????????????????????????????????????????????????????????C? Interpret the solution. Is it possible for Jed to burn 500 Calories per day by exercising for 45 minutes? If so, how?Yes; Jed should spend about 8 minutes walking and about 37 minutes running.???????????????????????????????????????????????????????????????????????????????REFLECT??3a. What is the minimum amount of time that Jed must exercise to burn 500 Calories per day? Explain.About 40.3 minutes; dividing 500 Calories by 12.42 Calories per minute (the???????????????????????????????????????????????????????????????????????????????????????rate of burn while running) gives about 40.3 minutes.??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????3b. What is the maximum amount of time that Jed must exercise to burn 500 Calories per day? Explain.About 97.7 minutes; dividing 500 Calories by 5.12 Calories per minute (the rate?????????????????????????????????????????????????????????????????????????????????of burn while walking) gives about 97.7 minutes.?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????3c. If the total number of Calories burned is held constant at 500, what happens to the amounts of time spent walking and running as exercise time increases from the minimum to the maximum?The amount of time spent walking increases while the amount of time spent????????????????????????????????????????????????????????????????????????????????????????running decreases.????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ................
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