A manufacturer produces x items per week at a total cost of Rs (x^2+78 x+2500 ) . The price per unit is given by 8 x=600- p where ' p ' is the price of each unit. Then the maximum profit obtained is

Step-by-step solution

Let x be the number of items produced per week. The total cost function is C(x) = x^2 + 78x + 2500 . From the equation 8x = 600 - p , the price per unit is p = 600 - 8x . The total revenue function is R(x) = p x = (600 - 8x)x = 600x - 8x^2 , and profit is P(x) = R(x) - C(x) = (600x - 8x^2) - (x^2 + 78x + 2500) . Simplifying, P(x) = -9x^2 + 522x - 2500 . Since the profit function is a downward-opening parabola, maximum occurs at the vertex with x = - b 2a = - 522 2 (-9) = 29 . Substituting x = 29 into P(x) gives P(2