$(10$ points$)$ Suppose the supply function for a certain item is given by $S(x)=x^2+10x,$ and the demand function is given by $D(x)=900-20x-x^2,$ determine the following:

a$)$ Find the equilibrium point.

$1$a$)$

b$)$ Find the consumer surplus at the equilibrium point.

$1$b$)$

c$)$ Find the producer surplus at the equilibrium point.

$1$c$)$

Let $p=D(x)$ be the consumer's demand function. The consumer's surplus is defined for the equilibrium point $(q^{**},p^{**})$ as:

Consumer Surplus $=CS={\displaystyle \underset{0}{\overset{q^{**}}{}}}D(x)dx-p^{**}{q}^{**}$

Let $p=S(x)$ be the producer's supply function. The producer's surplus is defined for the equilibrium point $(q^{**},p^{**})$ as:

Producer's Surplus $=PS=p^{**}{q}^{**}-{\displaystyle \underset{0}{\overset{q^{**}}{}}}S(x)dx$