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1 Algebra More applications of quadratic functions Name: There are many applications of quadratic functions in the real world. We have already considered applications for which we were given formulas and asked to find minimum or maximum values or x-intercepts. Today, we will consider two additional types of problems, projectile motion problems and problems involving geometric shapes. Projectile motion: Projectile motion involves objects that are dropped, thrown straight up, or thrown straight down. Factors that influence the motion of such an object include the height from which the object is dropped or thrown, the initial velocity of the object, and the downward pull of gravity on the object. The general formula for the height of a freely falling object is where 1 h a v0 h0 h is the height of the object after t seconds, t is the time in seconds, a is the acceleration due to gravity, v is the initial velocity of the object, and 0 h is the initial height of the object 0 Using -3 ft/s for a, the formula is Using -9.8 m/s, we obtain h 16 v0 h0 h 4.9 v0 h0

2 Example: An object is launched vertically upward from the top of a 100-foot tall tower at an initial velocity of 80 ft/s. a. When does the object reach its maximum height? b. What is the maximum height of the object? c. When will the object hit the ground? Solution: Because the units are feet and seconds, choose the formula v h( t) 16t v0t h0. Here, 0 and h 96, so the function that describes the height of the rocket is 0 0 ( ) h. a. The maximum height occurs at the vertex. Find the value of t at the vertex: b 0.5 a 3 seconds. Answer: The object reaches its maximum height.5 seconds after launch. b. Find the value of h at the vertex by subbing in.5 for t: h feet Answer: The maximum height of the object is 00 feet. c. When the rocket hits the ground, its height will be 0. Solve the equation 0 16t 80t 100 or 0 4t 0t 5 Solve using the quadratic formula: ( 4) Estimating: 0 0 or Answer: Because 0, the object will hit the ground after about 6.04 seconds.

3 Problems with geometric shapes: Examples: 1. The length of a rectangle is three more than twice the width. The area of the rectangle is 65 square meters. What are the dimensions of the rectangle? Solution: Let w = the width of the rectangle, ft w 3 = the length of the rectangle, ft Write the equation: lw A w 3 w 65 Solve by factoring: w 3w 65 0 w 10w 13w 65 0 w( w 5) 13( w 5) 0 ( w 5)(w 13) 0 w 5 0 w 5 or w 13 0 w 13 w 13 Because w 0, the width is 5 ft. Find the length: length w ft Answer: The width of the rectangle is 5 ft and the length is 13 ft.

4 . A garden measuring 1 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 85 square meters. What will be the width of the pathway? Solution: Let x = the width of the pathway Write expressions for the length and width of the garden with the pathway; length = 16 width = 1 Write an equation for the area of the garden and pathway: (x 16)(x 1) 85 4x 56x x 56x 93 0 Solve using the quadratic formula: (4) x 1.5 or x Answer: Because 0, the width of the pathway will be 1.5 ft.

5 Practice/Homework: 1. For a science competition, students must design a container that prevents an egg from breaking when dropped from a height of 50 feet. How long does the container take to hit the ground? Use the formula h 16 v h The height of a projectile launched vertically upward from the top of a 80-ft tall bridge is given by h 16 48t 80, where t is time in seconds. a. What is the maximum height of the projectile? b. How long will it take the projectile to strike the ground? 3. While playing catch with his grandson, Tim throws a ball into the air. The height in feet of the ball is given by h 16 64t 8, where t is time in seconds. How long will it take until the ball reaches his grandson s glove if he catches it at a height of 3 ft?

6 4. The length of a rectangular garden is feet less than twice the width. If the area of the garden is 40 square feet, what are the dimensions? 5. A city s skate park is a rectangle that is 100 feet long and 50 feet wide. The city wants to triple the area of the skate park by adding a border of width x to the length and width of the current park. What will be the dimensions of the new park? 6. The length of a rectangle is initially twice the width. Then 7 inches are added to both the width and the length. The area of the new larger rectangle is 130 square inches. Find the dimensions of the original rectangle.

7 7. A manufacturer of light fixtures has daily productions costs of C( x) x, where C is the total cost in dollars and x is the number of fixtures produced. a. What number of fixtures will result in the minimum daily cost? b. What is the minimum daily cost? 8. The value of Jennifer s stock portfolio is given by in hundreds of dollars and t is the time in months. v( t) 50 73t 3t, where v is the value of a. How much money did Jennifer start with? b. When will the value of Jennifer s portfolio be at a maximum? Answers: 1. It will take about 1.77 seconds for the container to hit the ground.. a. The maximum height is 316 ft. b. It will take about 5.94 seconds to hit the ground. 3. It will take about 4.08 seconds for the ball to reach the grandson s glove. 4. The width of the rectangle is 15 feet and the length is 8 feet. 5. The width of the new park is 100 feet and the length is 150 feet. 6. The width of the original rectangle is 3 inches and the length is 6 inches. 7. a. The minimum daily cost occurs with 16 fixtures. b. The minimum daily cost is $ a. Jennifer starts with $5000. b. Jennifer s portfolio will be at a maximum at about 1. months.

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