The supply function for a product is given by p=q^2+300, and the demand function is given by p + q = 410. Find the equilibrium quantity and price.
to find q, use p=410-q, and solve
410-q = q^2+300
q = 10
so, p = 400
To find the equilibrium quantity and price, we need to set the supply function equal to the demand function and solve for q.
The supply function is p = q^2 + 300.
The demand function is p + q = 410.
Setting these two equations equal to each other, we get:
q^2 + 300 = 410 - q.
Rearranging the equation, we have:
q^2 + q - 110 = 0.
Now, we can solve this quadratic equation for q using factoring or the quadratic formula.
Factoring the left side of the equation, we find:
(q + 11)(q - 10) = 0.
Setting each factor equal to 0, we have two possible solutions:
q + 11 = 0 => q = -11,
q - 10 = 0 => q = 10.
Since the quantity, q, cannot be negative in this context, the equilibrium quantity is q = 10.
Now, we can substitute q = 10 into either the supply function or the demand function to find the equilibrium price.
Using the demand function, we have:
p + 10 = 410,
p = 400.
Therefore, the equilibrium quantity is 10 and the equilibrium price is 400.
To find the equilibrium quantity and price, we need to find the point where the supply and demand functions intersect. At this point, the quantity supplied equals the quantity demanded.
Given:
Supply function: p = q^2 + 300
Demand function: p + q = 410
We can solve these equations simultaneously to find the equilibrium quantity and price.
Step 1: Substitute the supply function into the demand function.
(q^2 + 300) + q = 410
Step 2: Simplify the equation.
q^2 + q + 300 = 410
Step 3: Rearrange the equation to form a quadratic equation.
q^2 + q + (300 - 410) = 0
Step 4: Simplify the equation.
q^2 + q - 110 = 0
Step 5: Solve the quadratic equation using any method (factoring, completing the square, or using the quadratic formula).
In this case, we will solve it using the quadratic formula:
q = (-b ± √(b^2 - 4ac)) / 2a
In the equation q^2 + q - 110 = 0, a = 1, b = 1, and c = -110.
Plugging these values into the quadratic formula, we get:
q = (-1 ± √(1^2 - 4(1)(-110))) / (2(1))
q = (-1 ± √(1 + 440)) / 2
q = (-1 ± √441) / 2
q = (-1 ± 21) / 2
Step 6: Solve for q.
Case 1: q = (-1 + 21) / 2 = 20 / 2 = 10
Case 2: q = (-1 - 21) / 2 = -22 / 2 = -11
Since the quantity cannot be negative, we discard the second case, q = -11.
Therefore, the equilibrium quantity (q) is 10.
Step 7: Find the equilibrium price (p) using either the demand or supply function.
Using the demand function p + q = 410, we can substitute the found value of q:
p + 10 = 410
p = 410 - 10
p = 400
Therefore, the equilibrium price (p) is 400.
So, the equilibrium quantity is 10 units and the equilibrium price is $400.