P=90-2q , c=10 +0.5q2 what is the profit maximizing level of out put and the unit price
To find the profit-maximizing level of output and the unit price, we need to determine the quantity and price at which total profit is maximized.
Profit is calculated by subtracting the total cost (C) from the total revenue (P x q).
Given the following equations:
P = 90 - 2q (where P represents the unit price and q represents the quantity)
C = 10 + 0.5q^2 (where C represents the total cost)
Let's calculate the profit equation:
Profit = (P x q) - C
Substituting the equations for P and C:
Profit = (90 - 2q)q - (10 + 0.5q^2)
Now, we need to find the derivative of the profit equation with respect to q and set it equal to zero to find the maximum point. This will provide the value of q that maximizes profit.
Let's differentiate the profit equation:
dProfit/dq = (90 - 2q) - 2q - 0.5(2q)
Simplifying it:
dProfit/dq = 90 - 4q - q
Setting it equal to zero:
90 - 4q - q = 0
Simplifying further:
90 - 5q = 0
Now, solve for q:
5q = 90
q = 90/5
q = 18
So, the profit-maximizing level of output is q = 18.
To find the unit price (P) at this level of output, substitute the value of q in the equation for P:
P = 90 - 2q
P = 90 - 2(18)
P = 90 - 36
P = 54
Therefore, the profit-maximizing level of output is 18 units and the unit price is 54.