dt Thus the enclosed area is increasing at the rate of 80 cm s when dy Note is positive if y increases as x ncreases and negative if y decreases dx as x increases Example 4 The length x of a rectangle is decreasing at the rate of 3 cm minute and the width y is increasing at the rate of 2cm minute When x 10cm and y 6cm find the rates of change of a the perimeter and b the area of the rectangle Solution Since the length x is decreasing and the width y is increasing with respect to time we have and a The perimeter P of a rectangle is given by P 2 x y dP dt b The area A of the rectangle is given by A x y 150 dx dt Therefore When not be MATHEMATICS 2 dA dt dx dy dt dt dy dt Marginal Revenue When dx dt 3 6 10 2 as x 10 cm and y 6 cm 2 cm min y x Rationalised 2023 24 cost MC 2 cm min dy dt Example 5 The total cost C x in Rupees associated with the production of x units of an item is given by 2 3 2 2 cm min C x 0 005x 3 0 02 x 30x 5000 Find the marginal cost when 3 units are produced where by marginal cost we mean the instantaneous rate of change of total cost at any level of output dC dx Solution Since marginal cost is the rate of change of total cost with respect to the output we have Marginal 10 cm 0 005 3x 0 02 2x 30 x 3 MC 0 015 3 0 04 3 30 0 135 0 12 30 30 015 Hence the required marginal cost is 30 02 nearly Example 6 The total revenue in Rupees received from the sale of x units of a product is given by R x 3x 36x 5 Find the marginal revenue when x 5 where by marginal revenue we mean the rate of change of total revenue with respect to the number of items sold at an instant p rd Solution Since marginal revenue is the rate of change of total revenue with respect to the number of units sold we have dR dx x 5 MR 6 5 36 66 MR 6x 36