$D(x)$ 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.
\[
D(x)=1500-10 x, S(x)=900+5 x
\]
(a) What are the coordinates of the equilibrium point?
(b) What is the consumer surplus at the equilibrium point?
(c) What is the producer surplus at the equilibrium point?

Step 1 ：Set the demand function equal to the supply function to find the equilibrium quantity: \(1500 - 10x = 900 + 5x\)

Step 2 ：Solve the equation to find \(x = 40\)

Step 3 ：Substitute \(x = 40\) into either the demand or supply function to find the equilibrium price. For example, substituting into the demand function gives \(D(40) = 1500 - 10*40 = 1100\)

Step 4 ：So, the coordinates of the equilibrium point are \((40, 1100)\)

Step 5 ：Calculate the consumer surplus as the integral of the demand function from 0 to the quantity sold, minus the total revenue (price times quantity). This gives \(\int_0^{40} (1500 - 10x) dx - 1100*40 = 8000\)

Step 6 ：Calculate the producer surplus as the total revenue minus the integral of the supply function from 0 to the quantity sold. This gives \(1100*40 - \int_0^{40} (900 + 5x) dx = 4000\)

Step 7 ：\(\boxed{\text{(a) The coordinates of the equilibrium point are }(40, 1100)}\)

Step 8 ：\(\boxed{\text{(b) The consumer surplus at the equilibrium point is }8000}\)

Step 9 ：\(\boxed{\text{(c) The producer surplus at the equilibrium point is }4000}\)