$D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
\[
D(x)=\frac{169}{\sqrt{x}}, S(x)=\sqrt{x}
\]
(a) What are the coordinates of the equilibrium point?
(Type an ordered pair.)
(b) What is the consumer surplus at the equilibrium point?
$\$$ (Round to the nearest cent as needed.)
(c) What is the producer surplus at the equilibrium point?
(Round to the nearest cent as needed.)
Answer
So, the final answers are: (a) The coordinates of the equilibrium point are \((169, 13)\). (b) The consumer surplus at the equilibrium point is \(\boxed{2197}\). (c) The producer surplus at the equilibrium point is \(\boxed{732.33}\).
Step 1 ：First, we need to find the equilibrium point. This is where the demand equals the supply, i.e., \(D(x) = S(x)\). We can solve this equation to find the equilibrium quantity \(x\). Then we can substitute this quantity into either \(D(x)\) or \(S(x)\) to find the equilibrium price. In this case, we find that \(x = 169\) and the equilibrium price is \(13\).
Step 2 ：Next, we calculate the consumer surplus at the equilibrium point. This is the area between the demand curve and the price level, up to the equilibrium quantity. It can be calculated as the integral of the demand function from 0 to the equilibrium quantity, minus the total amount paid by consumers (which is the equilibrium price times the equilibrium quantity). In this case, the consumer surplus is \(2197\).
Step 3 ：Finally, we calculate the producer surplus at the equilibrium point. This is the area between the price level and the supply curve, up to the equilibrium quantity. It can be calculated as the total revenue received by producers (which is the equilibrium price times the equilibrium quantity), minus the integral of the supply function from 0 to the equilibrium quantity. In this case, the producer surplus is \(2197/3\) or approximately \(732.33\).
Step 4 ：So, the final answers are: (a) The coordinates of the equilibrium point are \((169, 13)\). (b) The consumer surplus at the equilibrium point is \(\boxed{2197}\). (c) The producer surplus at the equilibrium point is \(\boxed{732.33}\).
