D(x) is the price, in dollars per unit, that consumers are...

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. \(D(x) = (x-9)^2\), \(S(x) = x^2 + 4x + 15\)

Transcript

00:01 Hello everyone, in this problem we are given with the d of x which is the price in dollars per unit as x -9 the whole square and s of x is the price in dollars per unit as x square plus 4x plus 15.

00:23 So, now in the first part we need to find the equilibrium point.

00:27 So, for that we need to equate these two values demand and this is equal to s of x.

00:38 So, equating that x -9 the whole square will be equal to x square plus 4x plus 15.

00:47 So, now expanding this it should become x square plus 81 minus 18x to be equal to x square plus 4x plus 15.

00:57 So, now solving for x we have this to be x square plus 4x plus 15 minus x square minus 81 plus 18x to be equal to 0.

01:12 So, simplifying this we get this to be 22x minus 66 to be equal to 0.

01:18 So, from this we get 22x to be equal to 66 and solving for x we have the value of x to be 3.

01:28 So, we have the required equilibrium point.

01:33 So, now we need to find the coordinates for the equilibrium point.

01:38 So, for that we need to substitute the value of x to be 3 in s of x.

01:45 So, s of 3 will be equal to 3 square plus 4 multiplied by 3 plus 15.

01:53 So, simplifying this 9 plus 12 plus 15 we have this value to be 36.

01:59 So, therefore, the coordinates of equilibrium point is of 3 36.

02:22 So, now let us move on to the next part of the problem where we need to find the consumer surplus at the equilibrium point.

02:30 So, it is consumer surplus...

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