Question

In a factory, for production of Q articles, standing charges are ₹500, labour charges are ₹700 and processing charges are 50Q. The price of an article is 1700 - 3Q. Complete the following activity to find the values of Q for which the profit is increasing.

Solution: Let C be the cost of production of Q articles.

Then C = standing charges + labour charges + processing charges

∴ C = `square` 

Revenue R = P·Q = (1700 - 3Q)Q = 1700Q- 3Q²

Profit `pi = R - C = square`

 Differentiating w.r.t. Q, we get

`(dpi)/(dQ) = square`

If profit is increasing , then `(dpi)/(dQ) >0`

∴ `Q < square` 

Hence, profit is increasing for `Q < square` 

Answer 1

Let C be the cost of production of Q articles.

Then C = standing charges + labour charges + processing charges

∴ C = 1200 + 50Q

Revenue R = P·Q = (1700 - 3Q)Q = 1700Q- 3Q²

Profit `pi = R - C 

= (1700Q - 3Q² ) - (1200 + 50Q)

= 1700Q - 3Q² - 1200 - 50Q

∴  `pi` = 1650Q - 3Q² - 1200

 Differentiating w.r.t. Q, we get

`(dpi)/(dQ) = d/(dQ)(1650Q - 3Q^2 - 1200)`

`= 1650 xx 1 - 3 xx 2Q -0`

`(dpi)/(dQ) = ` 1650 - 6Q

If profit is increasing , then `(dpi)/(dQ) >0`

∴ 1650 - 6Q > 0

∴ 1650 > 6Q

∴ `Q <` 275 

Hence, profit is increasing for `Q <` 275