Find consumer's surplus at the market equilibrium point given that the demand function is p = sqrt(324−60x) and the supply function is p = x+9

Answer 1

the word "equilibrium" suggests they would be equal, so

√(324-60x) = x+9
square both sides
324 - 60x = x^2 + 18x + 81
x^2 + 78x - 243 = 0
(x - 3)(x + 81) = 0

x = 3 or x = -81

Don't know too much about market equilibriums, demand functions or supply functions, so do whatever you know with the above result.
I doubt if the negative x value is of any use.

( notice when x = 3, we get √144 = 12 , which is true
and we get √5184 = -72 which is false )

Answer 2

I looked this up, and it appears that the consumer surplus here would be15.
The equilibrium shows that the product is available at a price of 3, but the consumer is willing to pay 18 for it (demand p(0)).

@Nick, see whether I have interpreted this correctly.

Answer 3

Actually, I didn't finish reading the article cited below.
I just gave you the ∆P. The formula says that the consumer surplus is

c = ∫[0 .. equilibrium demand] (max price - equilibrium price) dq
In this case, that would be
∫[0,3] (√(324-60x) - 12) dq = 48/5 = 9.6

https://corporatefinanceinstitute.com/resources/knowledge/economics/consumer-surplus-formula/