The demand x of a commodity in terms of price p is given by x = 1 9 ( 245 - 7 p ) . The average cost is R s . 20 per unit. Find the profit function in terms of p .

Average cost = R s 20 Average cost = Total cost Number of units Total cost = Average cost × Number of units 20 × 1 9 ( 245 - 7 p ) = C ( x ) C ( x ) = 20 9 ( 245 - 7 p ) Demand function = 1 9 ( 245 - 7 p ) Revenue function = price × demand function R ( x ) = p × 1 9 ( 245 - 7 p ) = p 9 ( 245 - 7 p ) Profit function P ( x ) = R ( x ) - C ( x ) P ( x ) = p 2 ( 245 - 7 p ) - 20 2 ( 245 - 7 p ) = ( p - 20 ) 2 ( 245 - 7 p ) Hence, profit function is 1 2 ( 245 - 7 p ) ( p - 20 ) .

The demand x of a commodity in terms of price p is given by x = 1 9 ( 245 - 7 p ) . The average cost is R s . 20 per unit. Find the profit function in terms of p .

Average cost = R s 20 Average cost = Total cost Number of units Total cost = Average cost × Number of units 20 × 1 9 ( 245 - 7 p ) = C ( x ) C ( x ) = 20 9 ( 245 - 7 p ) Demand function = 1 9 ( 245 - 7 p ) Revenue function = price × demand function R ( x ) = p × 1 9 ( 245 - 7 p ) = p 9 ( 245 - 7 p ) Profit function P ( x ) = R ( x ) - C ( x ) P ( x ) = p 2 ( 245 - 7 p ) - 20 2 ( 245 - 7 p ) = ( p - 20 ) 2 ( 245 - 7 p ) Hence, profit function is 1 2 ( 245 - 7 p ) ( p - 20 ) .

The demand x of a commodity in terms of price p is given by x = 1 9 ( 245 - 7 p ) . The average cost is R s 20 per unit. Find the price and level of output at which profit is zero.

Average cost = R s 20 Average cost = Total cost Number of units Total cost = Average cost × Number of units 20 × 1 9 ( 245 - 7 p ) = C ( x ) C ( x ) = 20 9 ( 245 - 7 p ) Demand function = 1 9 ( 245 - 7 p ) Revenue function = price × demand function R ( x ) = p × 1 9 ( 245 - 7 p ) = p 9 ( 245 - 7 p ) Profit function P ( x ) = R ( x ) - C ( x ) P ( x ) = p 2 ( 245 - 7 p ) - 20 2 ( 245 - 7 p ) = 1 2 ( 245 - 7 p ) ( p - 20 ) When profit is zero, 1 2 ( 245 - 7 p ) ( p - 20 ) = 0 ( 245 - 7 p ) = 0 or ( p - 20 ) = 0 p = 35 or 20 Therefore, p = 20 Substituting in x = 1 9 ( 245 - 7 p ) = 1 9 ( 245 - 7 × 20 ) = 105 9 Hence, p = 20 , x = 105 9 .

Given the total cost function for x units of a commodity as C ( x ) = 1 3 x 3 + 3 x 2 - 16 x + 2 . Find the marginal cost function

Given, total cost function C ( x ) = 1 3 x 3 + 3 x 2 - 16 x + 2 Marginal cost function M C = d d x ( C ( x ) ) M C = d d x ( 1 3 x 3 + 3 x 2 - 16 x + 2 ) ⇒ M C = 1 3 × 3 x 2 + 3 × 2 x - 16 + 0 ⇒ M C = x 2 + 6 x - 16 Hence, marginal cost function is x 2 + 6 x - 16 .

Given the total cost function for x units of a commodity as C ( x ) = 1 3 x 3 + 3 x 2 - 16 x + 2 . Find the average cost function.

Given, total cost function C = 1 3 x 3 + 3 x 2 - 16 x + 2 Average cost function A C F = C Output A C F = 1 3 x 3 + 3 x 2 - 16 x + 2 x A C F = 1 3 x 2 + 3 x - 16 + 2 x Hence, average cost function is 1 3 x 2 + 3 x - 16 + 2 x .