A school is organising an art and craft exhibition. The management has decided to donate the profit…
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
Answer
AIWritten by AI (gemini-2.5-pro) - it can contain mistakes.
Total cost function: $C(x) = -x^2 + 11x + 50$.
Selling price per item $= ₹6 \implies R(x) = 6x$.
Profit function: $P(x) = R(x) - C(x) = 6x - (-x^2 + 11x + 50) = x^2 - 5x - 50$.
To earn a profit, $P(x) > 0$:
$x^2 - 5x - 50 > 0 \implies (x - 10)(x + 5) > 0$.
Since $x > 0$ (number of items), $x + 5 > 0$, so $x - 10 > 0 \implies x > 10$.
Therefore, more than 10 items must be sold to earn a profit (i.e., at least 11 items).
From ISC 2026 Mathematics Paper 1, question 20(ii).