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More Applications of Quadratic Functions

More Applications of Quadratic Functions. More Applications of Quadratic Functions. Example 1: A farmer wants to create a rectangular pen in order to raise chickens. Because of the location of the pen, the fence on the north and south sides of the rectangle will

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More Applications of Quadratic Functions

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  1. More Applications of Quadratic Functions

  2. More Applications of Quadratic Functions Example 1: A farmer wants to create a rectangular pen in order to raise chickens. Because of the location of the pen, the fence on the north and south sides of the rectangle will cost $5 per meter to construct whereas the fence on the east and west sides will cost $20 per meter. If the farmer has $1000 to spend on the fence, find the dimensions of the fence in order to maximize the area of the rectangle.

  3. More Applications of Quadratic Functions y Solution: Let x represent the length of the east and west sides. Let y represent the length of the north and south sides. A = xy (1) 5(y + y) + 20(x + x) = 10y + 40x 10y + 40x = 1000 (2) N W x E x S y

  4. More Applications of Quadratic Functions A = xy (1) 10y + 40x = 1000 (2) From (2) 10y = 1000 – 40x y = 100 – 4x  sub into (1) A = x(100 – 4x) A(x) = -4x2 + 100x  put into function notation

  5. More Applications of Quadratic Functions A(x) = - 4x2 + 100x  a = -4 b = 100 c = 0 The maximum area is 625 m2. This happens when x = 12.5 m and y = 100 – 4x = 100 – 4(12.5) = 50 m

  6. More Applications of Quadratic Functions Example 2: From the top of a 500 m cliff that borders the ocean, a cannonball is shot out horizontally and splashes down 2000 m from the base of the cliff. • Find the equation of the height, y, of the cannonball as a function of the horizontal distance, x, that the cannonball has traveled. • Determine the height of the cannonball when it is 1000 m away (horizontally) from the cliff.

  7. More Applications of Quadratic Functions Solution: a) Let the equation of the flight path be y = a(x – p)2 + q. Since the cannonball is shot out horizontally from the top of the cliff, the vertex of the flight path is (0, 500). So, y = a(x – 0)2 + 500 or y = ax2 + 500

  8. More Applications of Quadratic Functions Since the point (2000, 0) is on the flight path; y = ax2 + 500  0 = a(2000)2 + 500 - 500 = 4000000a Thus, the equation of the height in terms of the horizontal distance traveled is y = -0.000125x2 + 500

  9. More Applications of Quadratic Functions b) When the cannonball is 1000m away (horizontally), x = 1000, and thus; y = -0.000125x2 + 500 y = -0.000125(1000)2 + 500 y = -0.000125(1000000) + 500 y = 375 m Thus, the cannonball is 375 m above the ocean when it has traveled a horizontal distance of 1000m.

  10. Homework • Do # 3, 4, and 9 on pages 101 and 102 for Tuesday  • Don’t forget to study for your test 

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