1. More Applications of Quadratic Functions

2. 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 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) x x y y N E S W

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) = -4x 2 + 100x  put into function notation

5. More Applications of Quadratic Functions A(x) = - 4x 2 + 100x  a = -4 b = 100 c = 0 The maximum area is 625 m 2. 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. a) Find the equation of the height, y, of the cannonball as a function of the horizontal distance, x, that the cannonball has traveled. a) 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 + 500or y = ax 2 + 500

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

9. More Applications of Quadratic Functions b) When the cannonball is 1000m away (horizontally), x = 1000, and thus; y = -0.000125x 2 + 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