In this problem,p is in dollars and x is the number of units. If the demand function for a product is p = 28/(x + 1) and the supply function is p = 1 + 0.2x, find the consumer's surplus under pure competition. (Round your answer to two decimal places.

Answer :

Given the demand function :

[tex]p=\frac{28}{x+1}[/tex]

And the supply function:

[tex]p=1+0.2x[/tex]

So,

[tex]\begin{gathered} \frac{28}{x+1}=1+0.2x \\ \\ (1+0.2x)\cdot(x+1)=28 \\ 1\cdot(x+1)+0.2x\cdot(x+1)=28 \\ x+1+0.2x^2+0.2x=28 \\ 0.2x^2+1.2x+1=28 \\ 0.2x^2+1.2x-27=0 \end{gathered}[/tex]

using the general rule to solve the quadratic equation:

a = 0.2 , b = 1.2 , c = -27

so,

[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}=\frac{-1.2\pm\sqrt[]{1.2^2-4\cdot0.2\cdot(-27)}}{2\cdot0.2}[/tex][tex]\begin{gathered} x=\frac{-1.2\pm\sqrt[]{23.04}}{0.4} \\ \\ x=\frac{-1.2\pm4.8}{0.4} \\ \\ x=\frac{-1.2+4.8}{0.4}=\frac{3.6}{0.4}=9 \\ OR \\ x=\frac{-1.2-4.8}{0.4}=-15 \end{gathered}[/tex]

So, the answer is x = 9