Answer: [tex]\$11.5[/tex]
Step-by-step explanation:
Given
Demand function is [tex]p=22-2q[/tex]
The average cost function is [tex]c=2+q[/tex]
Total revenue is the product of demand and the price per unit.
[tex]r=\left(22-2q\right)q[/tex]
Profit is given by the difference of the total revenue and the cost
[tex]\Rightarrow P=r-c\\\Rightarrow P=22q-2q^2-2-q\\\Rightarrow P=-2q^2+21q-2[/tex]
Find the derivative of profit to get the maximum profit
[tex]\Rightarrow P'=-4q+21\\\text{Put the derivative equal to 0 to get the maximum profit}\\\\\Rightarrow q=\dfrac{21}{4}[/tex]
Put q in the equation of demand to get the price
[tex]\Rightarrow p=22-2\times \dfrac{21}{4}\\\\\Rightarrow p=22-10.5\\\Rightarrow p=\$11.5[/tex]
