Question

A school is organising an art and craft exhibition. The management has decided to donate the profit earned from the sale of exhibition items to an NGO.

• Total cost function for organising the exhibition is: `C(x) = −x^2 + 11x + 50`
• Each item is sold for 6.

Find the condition for the number of items to be sold to earn profit.

Answer 1

Identify Revenue and Profit Functions

Total Cost (C(x)): `x^2 + 11x + 50`

Total Revenue (R(x)): Since each item is sold for ₹6, R(x) = 6x

Profit (P(x)): Revenue − Cost

P(x) = 6x − (−x² + 11x + 50)

P(x) = 6x + x² − 11x − 50)

P(x) = x² − 5x − 50

To earn a profit, the profit function must be greater than zero (P(x)>0):

`x^2 − 5x − 50 > 0`

First, find the roots of the quadratic equation x2 − 5x − 50 = 0 by factoring:

(x − 10)(x + 5) = 0

The roots are x = 10 and x = −5

For the quadratic x² − 5x − 50 to be positive (> 0), x must be outside the roots:

x > 10 or x < −5

Since the number of items sold (x) cannot be negative, we ignore x < −5.

The condition to earn a profit is that the school must sell more than 10 items. (x > 10)