Recall that in business, a demand function expresses the quantity of a commodity demanded as a function of the commodity's unit price. A supply function expresses the quantity of a commodity supplied as a function of the commodity's unit price. When the quantity produced and supplied is equal to the quantity demanded, then we have what is called market equilibrium.
The demand function for a certain compact disc is given by the function p=-0.01x^2-0.3x+6 and the corresponding supply function is given by p=-0.01x^2-0.1x+2, where p is in dollars and x is in thousand of units.
The objective is to find the equilibrium point that is the price at which the quantity demanded equals the quantity supplied.
Setting the two equations equal to one another gives the quantity demanded equals the quantity supplied:
-0.01x^2 - 0.3x + 6 = -0.01x^2 - 0.1x + 2
This simplifies to:
0.20x = 4
Solving for x gives:
x = 4 / 0.20 = 20.
Therefore, the equilibrium quantity is 20 thousand units.
Substitute x = 20 into either the supply or the demand function to find the equilibrium price:
p = -0.01x^2 - 0.1x + 2
= -0.01 * 20^2 - 0.1 * 20 + 2
= -4 - 2 + 2
= -$4
Therefore, the equilibrium price is $4.
So, the market equilibrium is reached when 20,000 units are sold for a price of $4 each.
To find the market equilibrium, we need to set the quantity demanded equal to the quantity supplied and solve for the unit price (p).
Given:
Demand function: p = -0.01x^2 - 0.3x + 6
Supply function: p = -0.01x^2 - 0.1x + 2
First, let's set the quantity demanded equal to the quantity supplied:
-0.01x^2 - 0.3x + 6 = -0.01x^2 - 0.1x + 2
Now, we can simplify the equation by combining like terms:
-0.3x + 6 = -0.1x + 2
Next, let's isolate the variable by moving all terms with x to one side:
-0.3x + 0.1x = 2 - 6
-0.2x = -4
Now, divide both sides of the equation by -0.2 to solve for x:
x = (-4) / (-0.2)
x = 20
So, the quantity demanded and supplied is 20 thousand units.
To find the equilibrium price, substitute the value of x back into either the demand or supply function. Let's use the demand function:
p = -0.01(20)^2 - 0.3(20) + 6
p = -0.01(400) - 6 + 6
p = -4
Therefore, the market equilibrium for this compact disc is a quantity of 20 thousand units at a price of $4.
To find the market equilibrium, we need to determine the unit price at which the quantity demanded is equal to the quantity supplied.
First, let's find the quantity demanded by using the demand function:
Demand function: p = -0.01x^2 - 0.3x + 6
To find the quantity demanded, we set p (the unit price) equal to 0, and solve for x:
0 = -0.01x^2 - 0.3x + 6
This is a quadratic equation, so we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -0.01, b = -0.3, and c = 6. Plugging these values into the formula, we get:
x = (-(-0.3) ± √((-0.3)^2 - 4(-0.01)(6))) / (2(-0.01))
Simplifying:
x = (0.3 ± √(0.09 + 0.24)) / (-0.02)
x = (0.3 ± √0.33) / (-0.02)
Now we have two possible values for x. Let's calculate both:
x₁ = (0.3 + √0.33) / (-0.02)
x₂ = (0.3 - √0.33) / (-0.02)
Next, let's find the corresponding unit prices by plugging the values of x₁ and x₂ into the demand function:
p₁ = -0.01(x₁)^2 - 0.3(x₁) + 6
p₂ = -0.01(x₂)^2 - 0.3(x₂) + 6
Now let's find the quantity supplied using the supply function:
Supply function: p = -0.01x^2 - 0.1x + 2
Again, we set the unit price equal to 0 and solve for x:
0 = -0.01x^2 - 0.1x + 2
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = -0.01, b = -0.1, and c = 2. Substituting these values:
x = (-(-0.1) ± √((-0.1)^2 - 4(-0.01)(2))) / (2(-0.01))
Simplifying:
x = (0.1 ± √(0.01 + 0.08)) / (-0.02)
x = (0.1 ± √0.09) / (-0.02)
Again, we have two possible values for x. Let's calculate both:
x₃ = (0.1 + √0.09) / (-0.02)
x₄ = (0.1 - √0.09) / (-0.02)
Finally, let's find the corresponding unit prices by plugging the values of x₃ and x₄ into the supply function:
p₃ = -0.01(x₃)^2 - 0.1(x₃) + 2
p₄ = -0.01(x₄)^2 - 0.1(x₄) + 2
To find the market equilibrium, we need to find where the quantity demanded is equal to the quantity supplied. In other words, we need to compare the values of x₁, x₂, x₃, and x₄. If there is a match, then we have found the market equilibrium.