# Math 22 Business and Economics Applications

## Optimization in Business and Economics

1) Find the number of units ${\displaystyle x}$ that minimizes the average cost per unit ${\displaystyle {\overline {C}}}$ when ${\displaystyle C=2x^{2}+348x+7200}$

Solution:
Notice: ${\displaystyle {\overline {C}}={\frac {C}{x}}={\frac {2x^{2}+348x+7200}{x}}=2x+348+{\frac {7200}{x}}}$
Then, ${\displaystyle {\overline {C}}'=2-{\frac {7200}{x^{2}}}=0}$, so ${\displaystyle x^{2}=3600}$, so ${\displaystyle x=\pm {\sqrt {3600}}=\pm 60=60}$ since ${\displaystyle x}$ is positive.

2) Find the price that will maximize profit for the demand and cost functions, where ${\displaystyle p}$ is the price, ${\displaystyle x}$ is the number of units, and ${\displaystyle C}$ is the cost. Given the demand function ${\displaystyle p(x)=90-x}$ and the cost function ${\displaystyle C(x)=100+30x}$.

Solution:
Notice: The revenue function ${\displaystyle R(x)=x\cdot p(x)=x(90-x)=90x-x^{2}}$
The Profit function is ${\displaystyle P(x)=R(x)-C(x)=90x-x^{2}-(100+30x)=90x-x^{2}-100-30x=-x^{2}+60x-100}$
Then, ${\displaystyle P'(x)=-2x+60=0}$, so ${\displaystyle x=30}$
So, ${\displaystyle p(30)=90-30=60}$
Therefore, the price is ${\displaystyle \60}$ a unit will maximize the profit.