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