Question

how to calculate consumer surplus given q=20-0.05p^2 and p with a hyphen over it is 9

Answer 1

Is \(\bar{p}\) the price level?

Answer 2

yes @SithsAndGiggles that what 9 is

Answer 3

Okay, and does \(q\) denote the demand curve?

Answer 4

i believe so @SithsAndGiggles this is a demand equation I am asking about

Answer 5

Right. Then the consumer surplus is the area between the demand curve and the price level, which can be computed by the integral
\[\int_0^{\text{(wherever price and demand intersect)}}(20-0.05p^2)~dp\]

Answer 6

Oops, left out an important detail:
\[\int_0^{\text{(wherever price and demand intersect)}}(20-0.05p^2\color{red}{-9})~dp\]

Answer 7

I know that part because but im confused on how to find the intervals of where they meet? do you have to take the derivative of q to find the top interval for the integral? @SithsAndGiggles

Answer 8

No, it's just a matter of finding the intersection point. This occurs when both the price and demand curve are at the same point. The price level is fixed at \(9\), so the demand curve will equal \(9\) for a value of \(p\) that gives
\[9=20-0.05p^2\]
Solve for \(p\), and this will be the upper limit of the integral.

Answer 9

when I take the square root I get error @SithsAndGiggles

Answer 10

\[\begin{align*}
9&=20-0.05p^2\\
0.05p^2&=11\\
p^2&=220\\
p&=\sqrt{220}
\end{align*}\]
So you compute
\[\int_0^{\sqrt{220}}(20-0.05p^2-9)~dp\]