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Math 11 Applications Outcomes D6 and D7 APPLICATIONS WITH SUBSTITUTION Example 1: A rescue helicopter flies at 3 miles a minute. When the helicopter is 22 miles away a boat sends up a flare. The helicopter heads toward the boat. An equation for the distance d in miles away from the boat at time t minutes is d = -3t +22. The helicopter can spot a flare 6 miles away. The equation for the distance when the flare is visible is d = 6. How long until the helicopter pilot sees the flare? Example 2: A store receives an order of 300 TV sets in two deliveries. The first delivery has 60 more sets than the second delivery. The equations are: f + s = 300 and f = s + 60, where f is the number in the first delivery and s is the number in the second delivery. How many in each delivery? Math 11 Applications Outcomes D6 and D7 Example 3: The mass of wheat a farmer can supply is given by the equation W = 4000p – 3000 The mass of wheat that is bought (demand) is given by the equation W = -3000p + 25000 Where p is the selling price per bag and W is the mass of wheat. Determine the price per bag if supply equals demand Practice: For each question show the substitution and the working to obtain the answer. 1. A rescue helicopter flies at 5 miles a minute. When the helicopter is 35 miles away a boat sends up a flare. The helicopter heads toward the boat. An equation for the distance d in miles away from the boat at time t minutes is d = -5t +35. The helicopter can spot a flare 10 miles away. The equation for the distance when the flare is visible is d = 10. How long until the helicopter pilot sees the flare?1 2. A store receives an order of 1000 stereos in two deliveries. The first delivery has 250 more sets than the second delivery. The equations are: f + s = 1000 and f = s + 250, where f is the number in the first delivery and s is the number in the second delivery. How many in each delivery?2 1 2 10 hours 625 stereos in the first delivery and 375 stereos in the second delivery Math 11 Applications Outcomes D6 and D7 3. The mass of wheat a farmer can supply is given by the equation W = 3500p – 2000 The mass of wheat that is bought (demand) is given by the equation W = -4000p + 30000 Where p is the selling price per bag and W is the mass of wheat. Determine the price per bag if supply equals demand.3 4. An airplane cruises at a constant velocity of 840 km/h during the 5430-km trip from Montreal to London, England. The equation that describes its motion is d = -840t + 5430, where d is its distance from London in kilometers, and t is the time in hours since the airplane reached its cruising velocity. The plane must start its descent when it is 90 km from London, do d = 90. How long has the plane been cruising when it starts its descent?4 5. A car rental agency offers two rental plans for a certain class of car. Plan 1 is $60 per day with no charge for the number of kilometers driven. Plan 2 is $20 per day, plus $0.25 for each kilometer driven. You wish to rent a car for one day. The daily cost , c dollar, fro driving d kilometers is: Plan 1: C = 60 Plan 2: C = 0.25d + 20 a) Which plan is better for a one-day, 200-km trip? b) Solve the linear system. Determine how far you could drive so that the rental costs for each plan would be the same. What is the rental cost?5 3 $4.27/bag Approximately 6.36 h 5 a) Plan 1 b) 160 km; $60 4 Math 11 Applications Outcomes D6 and D7 6. Pyramid Stable scharges $20/h (including insurance) for trail rides. Sara’s Stables charges $16/h, with a separate fee of $12. Let the cost of the trail rides be C dollars and the number of hours be h. The equations relating the cost to hours for each stable are C = 20h and C = 16h + 12. a) How many hours of trail rides would result in the same total costs for each stable? b) Suppose you wish to go riding for 2 h. Which stable would you choose? Justify your answer.6 7. Ideally, the quantity of a crop that farmers harvest and sell at a given price should equal the quantity of the crop that consumers are willing to buy at that price. This is the law of supply and demand. The price of grapes is at least $1/kg and never more than $6.50/kg. The mass of grapes, G kilograms, that farmers will harvest (supply) is represented by G = 5000p − 5000, where p dollars is the selling price per kilogram. The quantity of grapes that consumers will purchase (demand) is represented by G = -4000p + 26 000. Determine the price per kilogram where the supply of grapes equals the demand.7 6 7 a) 3 h b) Pyramid Stables; answers will vary but work must support your answer Approximately $3.44/kg