50) the monthly demand and supply functions for the Luminar desk lamp are given by

p=d(x)= -1.1x^2 + 1.5x + 40
p= s(x)= 0.1x^2 + 0.5x + 15

respectively, where p is measured in dollars and x in units of a thousand. Find the equilibrium quantity and price.

deez n8ts

See similar example in:

http://www.jiskha.com/display.cgi?id=1307410080

To find the equilibrium quantity and price, we need to set the demand and supply functions equal to each other and solve for x.

Setting the demand and supply functions equal to each other:

-1.1x^2 + 1.5x + 40 = 0.1x^2 + 0.5x + 15

Combining like terms:

-1.1x^2 + 0.1x^2 + 1.5x - 0.5x + 40 - 15 = 0

Simplifying:

-1x^2 + 2x + 25 = 0

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this equation, a = -1, b = 2, and c = 25.

Plugging in the values:

x = (-2 ± √(2^2 - 4(-1)(25))) / (2(-1))

Simplifying:

x = (-2 ± √(4 + 100)) / (-2)

x = (-2 ± √104) / (-2)

Taking the square root of 104:

x = (-2 ± √(4 * 26)) / (-2)

x = (-2 ± 2√26) / (-2)

Simplifying:

x = 1 ± √26

Since we are looking for the equilibrium quantity, we can disregard the negative root:

x = 1 + √26

Now we can find the equilibrium price by substituting the value of x into either the demand or supply function. Let's use the demand function:

p = -1.1(1 + √26)^2 + 1.5(1 + √26) + 40

Calculating this expression will give us the equilibrium price of the Luminar desk lamp.

To find the equilibrium quantity and price, we need to set the demand and supply functions equal to each other and solve for x.

Setting the demand and supply functions equal, we have:

-1.1x^2 + 1.5x + 40 = 0.1x^2 + 0.5x + 15

To simplify the equation, let's move all terms to one side:

-1.1x^2 - 0.1x^2 + 1.5x - 0.5x + 40 - 15 = 0

Combining like terms:

-1.2x^2 + 1x + 25 = 0

This is a quadratic equation in standard form (ax^2 + bx + c = 0), where a = -1.2, b = 1, and c = 25. We can solve this quadratic equation by using the quadratic formula.

The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values a = -1.2, b = 1, and c = 25 into the quadratic formula:

x = (-(1) ± √((1)^2 - 4(-1.2)(25))) / (2(-1.2))

Simplifying further:

x = (-1 ± √(1 + 120)) / (-2.4)

x = (-1 ± √121) / (-2.4)

x = (-1 ± 11) / (-2.4)

Now, we have two possible solutions for x:

x₁ = (-1 + 11) / (-2.4) = 10 / (-2.4) ≈ -4.17

x₂ = (-1 - 11) / (-2.4) = -12 / (-2.4) = 5

Since the quantity cannot be negative, we only consider the positive value of x, which is x₂ = 5.

Now that we have the value of x, we can find the equilibrium price by substituting x = 5 into either the demand or supply function. Let's use the demand function:

p = -1.1x^2 + 1.5x + 40

Substituting x = 5:

p = -1.1(5)^2 + 1.5(5) + 40
p = -1.1(25) + 7.5 + 40
p = -27.5 + 7.5 + 40
p = 20

Therefore, the equilibrium quantity is 5 thousand units, and the equilibrium price is $20.