okay we got ourselves another math challenge here but we can solve it we can figure it out so the key piece of the regulation is that we want the power company to make be making zero economic profit which means we need to set the price equal to average cost what is average cost average cost is the total cost over the quantity produced what will the total cost be it'll be the variable cost plus the fixed costs what are the variable costs since we have a constant marginal cost we know what the total variable cost will be every unit of electricity costs us 5 cents to produce and so that means our variable cost is 5 cents times the quantity we produce and we're going to add to the variable cost the fixed cost of $1 million and then we're going to divide that whole thing by Q to make it the average cost so I can do some simplification here that this is going to mean the price is equal to 0.05 these Q's cancel out plus 1 million over Q so that's not too hard so we know that this needs to be true we have a price and a quantity two variables we need another equation with price and quantity so that we can solve for them and we have that with demand we want to find where demand now going to cross with this price so all we have to do is plug this in for the price so the quantity demanded will be 15 million I'm just writing in shorthand here minus 0.5 time 0.05 + 1 million over Q now I have q and Q I can solve for Q but we're going to find we're going to need the quadratic equation to do that so uh let's figure that out let's get this in the form that we need uh so we can multiply this through so we're going to have q is equal to 15 million minus 0.025 that's .5 * postive .05 that5 * positive 1 million over Q is going to be uh negative 500,000 over q and so we have that so these are two numbers so this is like one number here uh but then we have this Q in the denominator over here we have a Q over here it's getting kind of complicated but what I'm going to do is multiply everything by Q so we get q^2 is equal to 15 million - 0.025 * Q minus 500k because that Q while we multiply by Q it'll cancel this one out and now we can get it into our quadratic form and so we're going to have q^2 minus 15 million - 0.225 Q Plus 500 K is equal to zero and now we have that a is 1 uh B is uh this thing and C is 500k and so if you use the quadratic formula to do this calculation and you can find a calculator on the Internet or you could use wolf from alpha if you look at the problem above there's a worked out problem just like this one above there's a link to wolf from alpha that'll help you solve the equation at this stage so you won't even have to do these steps you could just do it right here but what I found is that Q is going to be equal the largest value for Q that we can get which is the one that we want is 14 million 999,999 N4 units of electricity and if we take this and plug it back in well we're going to plug it in really for Q here uh and solve for p we're going to get that the price that's being charged is 0.11 666 uh one more six in that so a little over 11 cents per unit of elect electricity but this is my answer that's the answer to the question is where do we want to set that price at we want to set it here that's where their uh price will be equal to their average cost of zero economic profit and they'll be producing the most electricity we can get them to produce uh viably so again uh you want to use the tools of calculation available to you you can solve at this stage for Q plug it in for P or plug it in to demand to find p and that's the answer uh so so it takes a bit of technology or I did it entirely on my calculator uh I recommend you do the same it's fun but there it is that's the answer