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^2-0.6x+10 and the corresponding supply function is given by p=-0.01x^2-0.2x+4, where p is in dollars and x is in thousand of units.
Find the equilibrium quantity and the corresponding price by solving the system consisting of the two given equations.
To find the equilibrium point, we need to equate the quantity demanded and the quantity supplied. This would mean setting the two equations equal to each other and solving for x:
-0.01x^2-0.6x+10 = -0.01x^2-0.2x+4
First, we can simplify by cancelling similar terms on both sides of the equation. The -0.01x^2 terms cancel out so we are left with:
-0.6x+10 = -0.2x+4
Next, we can rearrange the equation to isolate the variable:
-0.4x = -6
Finally, we can solve for x by dividing both sides of the equation by -0.4, thus getting our equilibrium quantity (x):
x = 15 (in thousands of units)
Now we can substitute x = 15 into either the demand or supply function to find the equilibrium price (p). Let's substitute it into the demand function:
p=-0.01(15)^2-0.6(15)+10
p=-0.01(225)-9+10
p=-2.25-9+10
p=-1.25+10
p=8.75
Therefore, the equilibrium quantity is 15,000 units and the equilibrium price is $8.75 per unit.
To find the equilibrium quantity and the corresponding price, we need to solve the system of equations formed by the demand and supply functions:
Demand function: p = -0.01x^2 - 0.6x + 10
Supply function: p = -0.01x^2 - 0.2x + 4
Setting the demand (p) equal to the supply (p), we get:
-0.01x^2 - 0.6x + 10 = -0.01x^2 - 0.2x + 4
To solve this equation, we can simplify by subtracting (-0.01x^2 - 0.2x + 4) from both sides:
-0.01x^2 - 0.6x + 10 - (-0.01x^2 - 0.2x + 4) = 0
Simplifying this further, we get:
-0.01x^2 - 0.6x + 10 + 0.01x^2 + 0.2x - 4 = 0
Combining like terms, we have:
-0.4x + 6 = 0
To solve for x, we can isolate x by subtracting 6 from both sides:
-0.4x = -6
Dividing both sides by -0.4 gives us:
x = -6 / -0.4
x = 15
Now that we have the value for x, we can substitute it back into either the demand or supply function to find the corresponding price. Let's use the demand function:
p = -0.01(15)^2 - 0.6(15) + 10
p = -0.01(225) - 9 + 10
p = -2.25 - 9 + 10
p = -2.25 + 1
p = -1.25
Therefore, at equilibrium, the quantity (x) is 15 thousand units, and the corresponding price (p) is $1.25.
To find the equilibrium quantity and corresponding price, we need to solve the system of equations consisting of the demand function and the supply function.
The demand function is given by:
p = -0.01x^2 - 0.6x + 10
The supply function is given by:
p = -0.01x^2 - 0.2x + 4
To find the equilibrium quantity, we set the quantity demanded equal to the quantity supplied and solve for x.
-0.01x^2 - 0.6x + 10 = -0.01x^2 - 0.2x + 4
By subtracting (-0.01x^2 - 0.2x + 4) from both sides, we simplify the equation to:
-0.6x + 10 = -0.2x + 4
Next, we isolate the variable x by subtracting -0.2x and 4 from both sides:
-0.6x + 0.2x = 4 - 10
Simplifying further, we have:
-0.4x = -6
Dividing both sides by -0.4, we find:
x = 15
So, the equilibrium quantity is x = 15 thousand units.
To find the corresponding price, we substitute the value of x into either the demand or supply function. Let's use the demand function:
p = -0.01(15)^2 - 0.6(15) + 10
Simplifying the equation, we have:
p = -0.01(225) - 9 + 10
p = -2.25 - 9 + 10
p = -2.25 + 1
p = -1.25
Therefore, the corresponding price at equilibrium is p = -1.25 dollars.
Note: The negative sign on the price indicates that the price is below zero, which may not be realistic in this context. These calculations assume that the given functions accurately represent the demand and supply relationships.