If a supply curve is modeled by the equation $ p = 125 + 0.002x^2 $, find the producer surplus when the
selling price is $625.
The producer surplus is $\$ 166,666.67$
Applications of Integration
Okay, So the question asks what producer surplus will be when we're given a good that is priced at $625 and I will dash that line and then a supply curve that'll draw in here of P is equal to 1 25 plus 0.2 x squared. So as a quick refresher producer surplus is going to be this area shaded in green to get it, we're going to first find this point of intersection. Then we're going to solve four this blue area, then this red area and subtract the two, which will leave us with the green area. So first to find the point of intersection. So to do that, we just set 625 equal to 1 25 plus 0.2 x squared. And then we subtract. Getting that 500 is equal to 0.2 x weird. And then once we divide out by 0.2 we get that X squared is equal to 250,000 so x could be plus or minus 500. And if you look at this price versus quantity graph. You can see that we're considering the positive branch of this parabola. So the value of X that we're concerned about is positive. 500. So now that we've found the point of intersection, first we get the blue area, which is just the area of a rectangle. So we solved for that by simply taking 625 multiplying by 500 and getting 312,000 500. Then for the red, I'm going to take the integral from zero 2 500 of 1 25 close 0.2 x squared DX. So we're going to get 1 25 x evaluated at zero and 500 plus 0.2 over three x cubed taken from bureau 2 500 which is 62,000 500 plus a 3333. And then there is a 0.3 repeating so altogether we're going to get 3 12 500 minus stuff in red again, 62,500 for the first part and 83,333 only three for the second part. And when all is said is said and done. We're going to get 166,000 666 point 66 dot, dot dot and because we're talking about dollars, we're going to get 166,000 600 and 67 dollars, and this right here is the answer.