 Hello and welcome to the session. In this session we discuss the following question which says a company starts producing pair of shoes and finds that the production cost of each pair is $20 and the fixed expenditure of production is $1500. If each pair is sold for $30 Determine the cost function, revenue function, profit function, rate given point and determine the rate given point if the fixed cost for producing the same pair of shoes increases to $2400 and variable cost is estimated to 20% of the total revenue. Before we move on to the solution, let's discuss some functions related to business and economics. That would help us to solve this question. First we have the cost function. The cost function is given as capital C equal to capital CX which is equal to F plus VX where this X is the quantity produced of a certain commodity and the C is the total cost of producing the commodity. So from this we can say that the total cost is broken into two parts where this F is the fixed cost and this VX is the variable cost. Fixed costs are those which do not change with various levels of production whereas the variable cost keep on varying as the levels of production vary. The fixed cost is independent of X and the variable cost is directly proportional to X. So this is how we represent the cost function. Next we have a revenue function. Revenue is basically the amount of money which is received from the sales of the goods. The revenue function is given as R is equal to RX where this R is the total revenue which is collected by the seller when it sells the X quantities produced. So the revenue function RX is equal to P into X where this P is the selling price per unit of the commodity and this X is the quantity sold. Next function that we have is the profit function. The profit function given as PX is equal to the total revenue that is RX minus the total cost which is CX. Here also this X is the total number of units of the commodity produced. Next we have the break even point. A break even point is basically a point where there is no profit and no loss. So if the profit would be equal to 0 then in that case the revenue function and the cost function would be equal. So at the break even point revenue function is equal to the cost function. This is the key idea that we use in this question. Let us now proceed with the solution. In the question we are given that the production cost of one pair of shoes is $20 and we are also given the fixed expenditure of production of the shoes which is $1500 and each pair is sold for $30. So first let's determine the cost function. First of all we assume let X be the number of pairs of shoes produced. It's given that the production cost of one pair is $20. So from here we can say that the production cost would be equal to $20. From the key idea we have that the cost function CX is equal to F plus VX that is the fixed cost plus the variable cost. So for the first part we have the cost function CX is equal to F plus VX. Now the fixed cost is $1500. So this means that F is $1500 plus the variable cost which is $20. So the cost function CX is equal to $1500 plus $20X. This is the answer for the first part. Now in the next part we need to find out the revenue function and we know that the revenue function CX is equal to P into X where this P is the selling price per unit of the commodity and X is the quantity sold. Now in the question we have that the selling price of each pair is $30 so the selling price X pairs would be $30X and so the revenue function would be equal to 30X that is the number of commodities sold multiplied by the selling price of each commodity. Next we will find out the profit function and we know that the profit function is equal to RX minus CX that is revenue function minus the cost function. So next we have the profit function PX is equal to revenue function RX minus the cost function CX. So from here we have 30X minus 1500 minus 20X which gives us 10X minus 1500 as the profit function. Then next we need to find out the breakeven point which we know is a no profit no loss point that is at this point there is no profit and no loss and so at the breakeven point the revenue function RX is equal to the cost function CX. So next we have at breakeven point cost function is equal to the revenue function and so this means that 1500 plus 20X is equal to 30X which gives us 10X is equal to 1500 or X is equal to 150. Thus we have the breakeven point is X equal to 150 that is there will be no profit no loss on the production and sale of 150 pairs of shoes. Like the next part of the question we have to determine the breakeven point as we are given the fixed cost for producing the same pair of shoes increases to 2400 dollars and the variable cost is estimated to 20% of the total revenue. So here we have the fixed cost as 2400 dollars. Now we already have the total revenue 30X which we have obtained in the earlier parts. In the question it is given that the variable cost is now estimated to be 20% of the total revenue so it is 20% of 30X which gives us 6X as the variable cost. So the variable cost is 6X dollars. Now the cost function CX is equal to fixed cost plus the variable cost so CX is equal to 2400 plus 6X. Now we know that at the breakeven point the cost function is equal to the revenue function which means that 2400 plus 6X is equal to the revenue function which is 30X so further we get 24X is equal to 2400 and from here we get that X is equal to 100. So therefore in this case we can say that the breakeven point is at X equal to 100. That is in the given conditions there will be no profit no loss on the sale of 100 pair of shoes. So this completes the session hope you have understood the solution of this question.