4 4 quadratic-curves

4.3 Quadratic Curves4.3 Quadratic Curves

Quadratic Curves


• Supply, demand, cost, revenue, and profit functions can also be quadratic curves.

• Examples:

• Quadratic curves – plane curves with equations of the form

• These curves are called conic sections.

€

Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0€

S(x) = x 2 + 30x

€

D(x) = −x 2 −100x + 2,400


• Types: Circle, Ellipse, Parabola, Hyperbola

• For more information, read Section 3.2 of the book.


1. Given the supply function S(x) = x2 + 6x + 9 and demand function D(x) = x2 – 10x + 25, find the market equilibrium quantity and price.

€

S(x) = D(x)

€

x 2 + 6x + 9 = x 2 −10x + 25

€

x =1

€

S(1) =12 + 6(1) + 9 =16

€

. . m e quantity: 1 unit

€

. . m e price: Php16


7. Given the demand function yD(x + 4) = 400 and supply function 2yS – x – 38 = 0, find the market equilibrium quantity and price.

€

yS = yD

€

400

x + 4=

x + 38

2

€

800 = (x + 38)(x + 4)

€

yS (12) = 12+382 = 25

€

. . m e quantity: 12 units

€

. . m e price: Php25€

x 2 + 42x − 648 = 0

€

x =12€

yS =400

x + 4

€

yD =x + 38

2


9. Given the cost function C(x) = 1,250 + 20x and revenue function R(x) = x(50 – 0.1x), find the break-even quantity and price.

€

C(x) = R(x)

€

1,250 + 20x = 50x − 0.1x 2

€

C(50) = 2,250

€

. . b e points: (50,2250) and (250,6250)€

x 2 − 300x +12,500 = 0

€

x = 50 or x = 250 €

C(250) = 6,250


19. When a particular computer accessory is sold for x pesos per unit, manufacturers will supply units to local retailers. The local demand would be 60 – x units.

a)At what market price will the manufacturers' supply of the items be equal to the consumers' demand for them?

b)How many units will be sold at this price?

€

x 2

10


€

S(x) = D(x)

€

x 2

10= 60 − x

€

x 2 +10x − 600 = 0

€

x = 20

€

D(20) = 60 − 20 = Php40€

(x + 30)(x − 20) = 0


25. Assume that a company's cost and revenue functions are C(x) = 6x + 120 and R(x) = 3x2+48x, respectively.

• Find the break-even price and quantity.• Set up the profit function and use it to find the

profit when 120 units are manufactured and sold.



• Find the break-even price and quantity.

€

C(x) = R(x)

€

6x +120 = 3x 2 + 48x

€

3x 2 + 42x −120 = 0

€

x 2 +14x − 40 = 0

€

x = 2.43€

C(2.43) =134.58

€

. . b e quantity: 3 units

€

. . b e price: Php134.58



• Set up the profit function and use it to find the profit when 120 units are manufactured and sold.

€

P(x) = R(x) − C(x)

€

=(3x 2 + 48x) − (6x +120)

€

=3x 2 + 42x −120

€

P(120) = 3(120)2 + 42(120) −120 = Php48,120

4 4 quadratic-curves

Education