Final answer:
The equilibrium price is found by setting the quantity demanded equal to the quantity supplied. Algebraic manipulation of the supplied demand and supply functions shows the equilibrium price is $2 with a quantity of 12. This can be confirmed visually using graphical methods as well.
Step-by-step explanation:
In the context of the given supply and demand functions, to find the equilibrium price, we need to set the quantity demanded (Qd) equal to the quantity supplied (Qs).
According to the supplied information, the demand function is:
Qd = 16 - 2P2 + 5P
And the supply function is:
Qs = -10P
Setting Qd equal to Qs for equilibrium:
16 - 2P2 + 5P = -10P
By rearranging the terms:
16 - 2P2 + 15P = 0
Now, solving for P, the equilibrium price, through algebra, we obtain:
P = $2
At this price, the quantity supplied and demanded is 12, indicating we have correctly found the equilibrium.
Moreover, if you prefer visuals, you can graph these functions to see the intersection at the equilibrium price and quantity. The demand curve graphed is P = 8 - 0.5Qd and the supply curve is P = -0.4 + 0.2Qs. The graphical solution confirms the algebraic one, with both approaches showing that the price is $2 and quantity is 12 at equilibrium.