D(x) is the price, in dollars per unit, that consumers are willing to pay for x

Question

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) = (x - 9)^2, S(x) = x^2 + 2x + 21 What are the coordinates of the equilibrium point? Square (Type an ordered pair.) What is the consumer surplus at the equilibrium point? $ square (Round to the nearest cent as needed.) What is the producer surplus at the equilibrium point? $ square (Round to the nearest cent as needed.)

Explanation / Answer

given D(x)=(x-9)2=x2-18x+81 ,S(x)=x2+2x+21

a)for equilibrium D(x)=S(x)

x2-18x+81=x2+2x+21

-18x+81=2x+21

20x=60

x=3

p=D(3)=(3-9)2=36

equilirium point (x,p)=(3,36)

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b) consumer surplus =[0 to 3] (D(x) -36) dx

consumer surplus =[0 to 3] (((x-9)2) -36) dx

consumer surplus =[0 to 3] (((1/3)(x-9)3) -36x)

consumer surplus =(((1/3)(3-9)3) -36*0) -(((1/3)(0-9)3) -36*0)

consumer surplus =63.00 dollars

c) producer surplus = [0 to 3] (36-S(x)) dx

producer surplus = [0 to 3] (36-(x2+2x+21)) dx

producer surplus = [0 to 3] (15-x2-2x) dx

producer surplus = [0 to 3] (15x-(1/3)x3-x2)

producer surplus =(15*3-(1/3)33-32) -(15*0-(1/3)03-02)

producer surplus =27.00 dollars

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