The unit cost, in dollars, to produce bins of cat food is $3 and the fixed cost is $6972. The price-demand function, in dollars per bin, is p(x)=253−2xFind the cost function. C(x)=Find the revenue function. R(x)=Find the profit function. P(x)=At what quantity is the smallest break-even point?

Answer:Revenue , Cost and Profit FunctionStep-by-step explanation:Here we are given the Price/Demand Function as P(x) = 253-2xwhich means when the demand of Cat food is x units , the price will be fixed as 253-2x per unit.Now let us revenue generated from this demand i.e. x units Revenue = Demand * Price per unitR(x) = x * (253-2x)       = [tex]253x-2x^2[/tex]Now let us Evaluate the Cost Function Cost = Variable cost + Fixed Cost Variable cost = cost per unit * number of units                       = 3*x                       = 3xFixed Cost = 6972 as given in the problem. HenceCost Function C(x) = 3x+6972Let us now find the Profit Function Profit = Revenue - Cost P(x) = R(x) - C(x)= [tex]253x-2x^2 - (3x + 6972)[/tex]= [tex]253x-3x-2x^2-6972\\= 250x-2x^2-6972\\=-2x^2+250x-6972\\[/tex] Now we have to find the quantity at which we attain break even point. We know that at break even point Profit = 0 Hence P(x) = 0 [tex]-2x^2+250x-6972=0\\[/tex]now we have to solve the above equation for x Dividing both sides by -2 we get [tex]x^2-125x+3486=0[/tex]Now we have to find the factors of 3486 whose sum is 125. Which comes out to be 42 and 83 Hence we now solve the above quadratic equation using splitting the middle term method . Hence [tex]x^2-42x-83x+3486=0\\x(x-42)-83(x-42)=0\\(x-42)(x-83)=0\\[/tex]Either (x-42) = 0 or (x-83) = 0 therefore if x-42= 0 ; x=42if x-83=0 ; x=83Smallest of which is 42. Hence the number of units at which it attains the break even point is 42.