more applications of quadratic functions. example 1: a farmer wants to create a rectangular pen in...

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.


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


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


A(x) = - 4x2 + 100x a = -4 b = 100 c = 0

2

2

4,

2 4

100 4( 4)(0) (100),

2( 4) 4( 4)

100 0 10000,

8 16

25,625

2

b ac b

a a

The maximum area is 625 m2. This happens when x = 12.5 m and

y = 100 – 4x

= 100 – 4(12.5) = 50 m


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.


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


Since the point (2000, 0) is on the flight path;

y = ax2 + 500 0 = a(2000)2 + 500

- 500 = 4000000a500

40000001

80000.000125

a

a

a

Thus, the equation of the height in terms of the horizontal distance traveled is

y = -0.000125x2 + 500


b) When the cannonball is 1000m away(horizontally), x = 1000, and thus;

y = -0.000125x2 + 500y = -0.000125(1000)2 + 500y = -0.000125(1000000) + 500y = 375 m

Thus, the cannonball is 375 m above the ocean when it has traveled a horizontal distance of 1000m.

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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