Word Problems Involving Quadratic Functions 1. The Starks have 24 m of fencing with which to make a rectangular dog run. If they use a side of the house as one side of the run, what dimensions will give the maximum area?

2. A rectangle has a perimeter of 30 meters and an area of 54 square meters. Find its dimensions.

3. Find two numbers whose sum is 16 and whose product is a maximum.

4. Find two numbers that differ by 10 and have a product that is a minimum.

5. Cathy has 28 m of fencing to make a rectangular pen for some rabbits. What dimensions will give a maximum area?

6. A certain 120 V electrical circuit has a resistance of 12 amps. The power P (in watts) that can be produced in the circuit when a current i (amperes) is flowing is given by P = −12i 2 + 120i . Find the maximum power that can be produced in the circuit.

7. An object is launched at 19.6m/s from a 58.8 m tall platform. The equation, h = −4.9t 2 + 19.6t + 58.8 , represents the objects motion, where h represents the height (m) of the object from the ground and t represents the time (sec). When does the object hit the ground? When does it reach its maximum height.

8. Clay pigeons are fired upward from ground level, and the equation h = −4.9t 2 + 14.7t , represents the motion, where h represents height (m) and t represents time (s). What will the clay pigeons maximum height be? How high will the clay pigeon be after 3 seconds?

9. A rock is propelled from the edge of a cliff that is 2940 m above sea level. The equation, h = −4.9t 2 + 245t + 2490 , represents the rocks trip, where h represents height (m) and t represents time (s). After how many seconds will the rock hit the water below?

10. A baseball player throws a ball into the air. If the equation that represents the ball path is h = −2t 2 + 6t + 8 , where h represents height (ft) and t represents time (s). What is the height of the ball initially? What is the height of the ball after 1 sec? How long is the ball in the air?

11. A rocket was launched from a platform that is 500 ft high. The path of the rocket is represented by h = −4.9t 2 + 2450t + 500 , where h represents height (ft) and t represents time (sec). How long does it take the rocket to be 500 ft above the ground?

12. In 1900, the standing high jump was introduced as an event in the Olympic Games. Ray Ewry of the USA won the event with a jump of 1.65m. If he started his vertical jump at a horizontal distance of 0.30m from the crossbar, find the quadratic function that represents his jump. Predict the height of the jump at 0.1m from the bar.

13. A bridge is 100m long and has a maximum height of 40m. Sketch a graph of the bridge and find the quadratic equation that represents this bridge. If a boat that is 30m high sails underneath the bridge and is 20m from shore, will the boat hit the bridge or coast through?

14. At football games they sometimes shoot the teams logo t-shirts out of a big projectile gun into the bleachers. A shirt travels 20m and reaches a maximum height of 60m, if the gun is shot from 3m above the ground. If a person on the bleachers is 40m from the ground and catches the t-shirt, how far is he from the gun (on the horizontal)?

15. A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around increasing the total area to 285 square meters. What will be the width of the pathway?

Practice Problems: 1. The length of a rectangle is 10cm less than 4 times the width. If the perimeter is 130cm, find the dimensions of the rectangle. 2. A ball is thrown up into the air on a parabolic path given by the function y = −2 x 2 + 10 x , where t is the time in seconds after the ball is tossed and y is the ball’s height in meters. a. At what time will the stone be at maximum height? b. What is the maximum height? c. How high is the stone after 2 seconds? d. How long is the stone in the air? 3. A submarine traveling in a parabolic arc ascends to the surface. The path of the submarine is described by y = 2 x 2 − 10 x − 50 , where x represents the time in minutes and y represents the submarines depth in meters. a. How deep is the submarine initially? b. For how long is the submarine underwater? c. What maximum depth does it reach? 4. At a supervised beach, a lifeguard has used 620m of marker buoys to rope off a rectangular swimming area. If one side of the swimming area is the shoreline, calculate the dimensions of the swimming area so the area is a maximum. 5. A handball court is to be surrounded by plastic stripping 30m long. One side of the court does not need the stripping because it is a brick wall. What dimensions of the court will give the maximum area? 6. Two numbers, x and y, are related such that 3 x + y = 42 . Find the two numbers so that their product is a maximum. 7. A rectangular playing field is fenced into three sections as shown. If the total amount of fencing used is 800m, what are the values of x and y so that the area is a maximum? y

y

y

x

8. Two consecutive odd numbers are added to a third number to total 48. What are the three numbers so that the sum of their squares is a minimum? 9. Determine two numbers whose difference is 12 and whose product is a minimum. 10. The sum of two numbers is 20. What is the least possible sum of their squares?

11. A bullet fired vertically at 80m/s will have its height h, in meters, after time t, in seconds, given by

h = 80t − 5t 2 . What height will the bullet reach? How long will it take to reach this height? 12. A rectangular paddock is to be enclosed using only 500m of fencing. One side of the paddock borders the barn and does not require fencing. What is the maximum area of the paddock? What dimensions will give this maximum area? 13. The diagram shows a plan for a deck that is to be built on the corner of a cottage. A railing is to be constructed around the four outer edges of the deck. The two long sides are equal and the two short sides are equal. There is 30m of railing to be used. Find the maximum area of the deck? DECK

COTTAGE 2

14. The path of a thrown baseball can be modeled by the function h = −0.004d + 0.14d + 2 where h is the height of the ball, in meters, and d is the horizontal distance of the ball from the player, in meters. Find the maximum height of the ball. How far is the ball from the player when it reaches this height? 15. The sum of the squares of two consecutive positive numbers is 265. Find the integers. 16. A square garden is to be increased in size by extending the length by 12m and the width by 10m. The area of the new rectangular garden is 675m2. What are the dimensions of the garden? 17. The sum of the squares of two consecutive even integers is 580. Find the integers. 18. Find two consecutive positive odd numbers that have a product of 483. 19. A rectangular skating rink measuring 30m by 20m is doubled in area by adding a strip at one end of the rink, and a strip of the same width along one side of the rink. Find the width of the strips if the enlarged rink is still rectangular. 20. A chemical plant is rectangular and has a length of 100m and a width of 60m. A safety zone of uniform width surrounds the plant. If the area of the safety zone equals the area of the plant, what is the width of the safety zone? 21. When 15m is added to the length of two opposite sides of a square and 5 m is added to the length of the other sides. The area of the resulting rectangle is 441m2. Find the length of the sides of the original square. 22. The underside of a bridge has the shape of a parabolic arch. It has a maximum height of 30m and a width of 100m. Can a boat with a height of 20m and a width of 30m pass under the bridge? 23. A man working on a bridge has a life support rope tied to his waist. The bridge is 60m long with a maximum height of 20m from the water below. If the man is working 5m from the centre of the bridge, what would be the

maximum length the rope can be so that the man would not hit the water if he fell? 24. Find two numbers whose sum is 20 and whose product is a maximum. 25. The Russo’s fenced the rectangular yard at their lake cottage on three sides and used the lake as the fourth side. If 120 m of fence is used and the yard is of maximum area, what are the dimensions? 26. Find the number that, when added to its own square, will give a minimum sum. What is the sum? 27. An object is launched directly upward at 64 ft/s from a platform 80 ft high. The equation, h = −16t 2 + 64t + 80 , represents the objects motion, where h Æ height (ft) and t Æ time(s). What is the objects maximum height? At what time does the object reach its maximum height? 28. A book is thrown from a cliff at a height of 160 ft. The equation, h = -16t2 – 48t + 160, represents the books descent, where h Æ height (ft) and t Æ time (s). When does the book hit the ground? 29. Deep within a cave (shaped like a parabolic arch) a bat, 25cm in length, is hanging upside down on the underside of the cave, 1.75m from the highest point in the arch. If the cave is 8m wide and has a maximum height of 3m, how far is the bat from the floor of the cave? 30. A Frisbee is thrown straight up in the air from a position of 2 m above ground level. Because of the wind the Frisbee travels a path that is represented by the formula, h = 2 + 6t − 2t 2 , where h Æ height (m) and t Æ time (s). What is the maximum height the Frisbee will reach? If it is caught 2 m above the ground, how long will it have been in the air? 31. A playground, which measures 50m by 35m, is to be doubled in area by adding a strip of uniform width around the outside of the existing area. What is the width of the new strip around the playground? 32. A cliff diver jumps off a cliff that is 40 m high. If the equation h = -4.9t2 + 4.9t + 30 represents his dive, where h Æ height (m) and t Æ time (sec), when will he be 0.6 m from the water? 33. A jet traveled to Calgary from St. John’s, a distance of 4200km. On the return flight, a tail wind increased the speed of the jet by 100km/h. The total return time (to Calgary and back again) was 13 hours. What was the average speed of the jet from St. John’s to Calgary? 34. A droplet of water is ejected vertically by a geyser at a starting speed of 30 m/s. The approximate height that it will reach in t seconds is given by h = −5t 2 + 50t + 40 . What is the maximum height reached by the droplet?

ANSWERS: Height at 35m and 65m is 27.3 so that the boat at 20m will pass underneath Height where the bat is hanging is 2.43m so the bat’s head is 2.18m from the floor Max length of rope 19.4m