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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 functions1
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.

more applications of quadratic functions2
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

more applications of quadratic functions3
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

more applications of quadratic functions4
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

more applications of quadratic functions5
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.
more applications of quadratic functions6
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

more applications of quadratic functions7
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

more applications of quadratic functions8
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.

homework
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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