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Solving Quadratic Problems: Modeling Real-Life Scenarios with Parabolas

This guide explores practical applications of quadratic functions in real-world scenarios, specifically through modeling the height of projectiles like fireworks and the shape of structures like suspension bridges. We demonstrate how to derive equations for various parabolic paths, including the flight of a firework launched from a certain height, and the calculation of heights at defined distances. Furthermore, we analyze the design of parabolic arches in architecture by establishing equations that represent their shapes, providing a comprehensive approach to problem-solving using quadratics.

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Solving Quadratic Problems: Modeling Real-Life Scenarios with Parabolas

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  1. Solving realistic problems using quadratics The trick is selecting the BEST (quickest, easiest, laziest) way to solve!

  2. EX 1 This parabola models the height of a ball tossed into the air. Determine its equation.

  3. EX 2 At a fireworks display, a firework is launched from a height of 2 m above ground and reaches a maximum height of 40 m, 10 m away along the ground from where it was launched. • A) Determine an equation to model the flight path

  4. EX 2 At a fireworks display, a firework is launched from a height of 2 m above ground and reaches a maximum height of 40 m, 10 m away along the ground from where it was launched. • B) After the firework reaches its max height, it continues on its path for another 1 m in terms of horizontal distance before it explodes. What is its height when it explodes?

  5. EX 2 At a fireworks display, a firework is launched from a height of 2 m above ground and reaches a maximum height of 40 m, 10 m away along the ground from where it was launched. • C) In the last question, we found the firework is at a height of _______ where the horizontal distance from the launch point is ______. • At what other place (horizontally) is the firework at the same vertical height?

  6. EX 3 The Dufferin Gate is a parabolic arch that is approximately 20 m tall and 22 m wide. • Determine an equation to model the arch.

  7. EX 4 p. 278 #18: Many suspension bridges hang from cables that are supported by two towers. The shape of the hanging cables is very close to a parabola. A typical suspension bridge has large cables that are supported by two towers that are 20 m high and 80 m apart. The bridge surface is suspended from the large cables by many smaller vertical cables. The shortest vertical cable is 4 m long. • A) Using the bridge surface as the x-axis, find an equation to represent the parabolic shape of the large cables

  8. EX 4 p. 278 #18: Many suspension bridges hang from cables that are supported by two towers. The shape of the hanging cables is very close to a parabola. A typical suspension bridge has large cables that are supported by two towers that are 20 m high and 80 m apart. The bridge surface is suspended from the large cables by many smaller vertical cables. The shortest vertical cable is 4 m long. • B) How long are the vertical cables that are 25 m from each tower?

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