 Hello and welcome to the session. In this session we discuss the following question which says suppose the demand per month for commodity is 12 if the price is $15 and 8 if the price is $23 assuming that the demand curve is linear determine first the demand function, second the average revenue, third the marginal revenue, fourth the actual revenue from selling the 10th item. First of all let's see what is the demand function it is given as x is equal to f of p where this p is the price per unit and x is the number of units demanded at this price p only. Then next we have the revenue function which is given as r is equal to r of x is equal to p into x where again this p is the price per unit x is the number of units that are being sold and r is the total revenue elected by selling x units. Next we have the average revenue which is denoted by ar and this is the revenue received per unit so average revenue is equal to the total revenue which is r upon the number of units sold that is x and we know that r is equal to px so we have the average revenue ar is equal to px upon x so this comes out to be equal to p so we can say that average revenue is same as the price per unit. Next we have the marginal revenue which is denoted by mr and this is the rate of change of the total revenue with respect to the quantity sold which is dr by dx so as r is px so d by dx of px is mr and this means that the marginal revenue mr is equal to p plus x into dp by dx so 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 demand curve is linear so we can take let the linear demand function d equal to a plus b into x where this p is the price per unit and x is the number of units demanded and we are given that the demand per month for commodity is 12 if the price is 15 that is if the price of the commodity is 15 dollars per unit then the commodity demanded are 12 in number and if the price of the commodity is 23 dollars per unit then the commodity demanded is 8 so to find out the demand function we first consider x equal to 12 and in this case the price per unit that is p is 15 dollars and when we have x equal to 8 when the price per unit is 23 dollars now putting the values of x and p this equation say equation 1 we get 15 is equal to a plus 12 b and 23 is equal to a plus 8 b let this be equation 2 and this p equation 3 now we need to solve these two equations so solving equations 2 and 3 we get b as minus 2 value of a would be equal to 39 so now substituting the values of a and b in equation 1 that is in this equation we get p is equal to 39 minus 2x so this is our demand function now in the x part we need to find out the average revenue r upon x that is the revenue upon the number of units sold so for this first of all we will find out the revenue function which is given by r that is r of x and this is equal to p into x so r is equal to p into x which is 39 minus 2x the whole into x so we have r is equal to 39x minus 2x square now we know that the average revenue a r is equal to total revenue r upon x which is equal to 39x minus 2x square this whole upon x so we have average revenue a r is equal to 39 minus 2x this is the average revenue now next we will find the marginal revenue which is the rate of change of the revenue with respect to the quantity sold so in the next part the marginal revenue is equal to mr and this is equal to dr by dx thus mr is equal to b by dx of 39x minus 2x square the whole so further this is equal to 39 minus 4x this is the marginal revenue then in the next part we need to find the actual revenue from selling the 10th item so the actual revenue from selling the 10th item is equal to the revenue received on selling items minus the revenue received on selling nine items as we know that the total revenue is given by the function 39x minus 2x square so the revenue received on selling 10 items would be given by r at 10 that is by putting the value of x as 10 so 39 into 10 minus 2 into 10 square which is 100 now this whole minus the revenue received on selling nine items that is we now put the value of x as 9 here so it is 39 into 9 minus 2 into 9 square that is 81 the whole further this is equal to 390 minus 200 minus 39 into 9 is 351 minus 2 into 81 is 162 the whole so further we get this is equal to 390 minus 200 minus 351 plus 162 which is equal to 552 minus 551 and this is equal to 1 therefore we can say that the actual revenue received from selling the 10th item is given as 1 so this is the answer for the fourth part of the question so this completes the session hope you have understood the solution of this question