Respuesta :
The consumers’ surplus (CS) if the market price is set at $6 is 16.67 (per unit thousands).
Let,
The wholesale unit price in dollars = p
The quantity demanded each week = x (in thousand units)
The demand function = [tex]-0.01x^2-0.2x+9\\[/tex]
Market price = $6
Substitute p = 6 into the demand function:
6= -0.01[tex]x^2[/tex]-0.2x+9
0= -0.01[tex]x^2[/tex]-0.2x+3 (subtract 6 on both sides of the equation)
0.01[tex]x^2[/tex]+0.2x-3=0 (rewrite the equation)
[tex]x^2[/tex]+20x-300=0 (multiply 100 on both sides of the equation)
(x+30)(x-10)=0 (factor the equation)
(x+30)=0 and (x-10)=0 (zero product property of multiplication)
Therefore, x=-30, x=10.
x=10 as negative numbers are not feasible in this situation.
Consumer surplus (CS) of a production can be found using the integral,
CS = [tex]$ \int_{0}^{x}(p)dx $ -px[/tex]
CS = [tex]$ \int_{0}^{x} (-0.01x^2-0.2x+9)dx $ -px[/tex]
Substitute x=10 and p=6 :
CS = [tex]$ \int_{0}^{10} (-0.01x^2-0.2x+9)dx $ - (6\times10)[/tex]
CS = [tex]$ \int_{0}^{10} (-0.01x^2-0.2x+9)dx $ -60[/tex]
CS = [tex]$ \left[\frac{x^3}{300}-\frac{x^2}{10}+9x\right]_{0}^{10} $ -60[/tex](evaluate the integral)
CS = [tex](((10^3/300)-(10^2/10)+(9$ \times $10))-(0)) - 60[/tex] (evaluate the limit)
CS = (230/3)-60
CS = 50/3
CS [tex]$ \approx $[/tex] 16.67
Therefore, The consumers’ surplus (CS) is 16.67 if the market price is set at $6(per unit thousands).
Learn more about application of integrals at:
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