The supply and demand for a business manufacturing toy cars are given by these two equations, where y is the price in dollars per item and x is the number of toy cars sold in thousands.
Supply: y = x^2 − x + 0.5
Demand: y = x^2 − 1.6
The two graphs intersect at the point (2.1, 2.8), which is the market equilibrium for the business. Which statement is true?
A) The market will be in equilibrium when the price of each toy car is $2.10.
B) The market will be in equilibrium when 2,800 toy cars are sold.
C) The market will be in equilibrium when 2,100 toy cars are sold.
D) The market will be in equilibrium when 2,450 toy cars are sold and the price of each toy car is $2.10.
E) The market will be in equilibrium when 7.37 toy cars are sold and the price of each toy car is $2.12.
To determine which statement is true, we need to analyze the information provided.
First, let's recall what market equilibrium means. In a market equilibrium, the quantity supplied equals the quantity demanded, resulting in a balance between the sellers and buyers.
Given the supply equation, y = x^2 - x + 0.5, and the demand equation, y = x^2 - 1.6, we can find the equilibrium point by setting the supply equal to the demand:
x^2 - x + 0.5 = x^2 - 1.6
To solve this equation, we can subtract x^2 from both sides:
-x + 0.5 = -1.6
Next, we can isolate x by subtracting 0.5 from both sides:
-x = -2.1
Finally, we multiply both sides by -1 to obtain:
x = 2.1
Substituting x back into either the supply or demand equation, we can find the corresponding y-value:
y = 2.1^2 - 1.6
y = 4.41 - 1.6
y = 2.81
The equilibrium point is (2.1, 2.81).
Now, let's evaluate the statements:
A) The market will be in equilibrium when the price of each toy car is $2.10.
The price is not directly given in the equilibrium point, only the quantity.
B) The market will be in equilibrium when 2,800 toy cars are sold.
The quantity does not match the equilibrium point of 2.1 (thousand cars).
C) The market will be in equilibrium when 2,100 toy cars are sold.
The quantity does not match the equilibrium point of 2.1 (thousand cars).
D) The market will be in equilibrium when 2,450 toy cars are sold and the price of each toy car is $2.10.
The quantity does not match the equilibrium point of 2.1 (thousand cars), and the price is not directly given in the equilibrium point.
E) The market will be in equilibrium when 7.37 toy cars are sold and the price of each toy car is $2.12.
The quantity does not match the equilibrium point of 2.1 (thousand cars), and the price is close but does not match the price mentioned.
Therefore, the correct statement is:
A) The market will be in equilibrium when the price of each toy car is $2.10.