<p>A company manufactures cassettes. Its cost and revenue functions are \(C(x) = 26{,}000 + 30x\) and \(R(x) = 43x\), respectively, where \(x\) is the number of cassettes produced and sold in a week. How many cassettes must be sold by the company to realise some profit?</p>
Step-by-Step Solution
Key Concept: Profit occurs when Revenue exceeds Cost, so we need R(x) > C(x). Set up the inequality 43x > 26,000 + 30x and solve for the minimum x where profit becomes positive.
<p><strong>Step 1:</strong> Set up the profit condition. Profit P(x) = R(x) - C(x) > 0</p><p><strong>Step 2:</strong> Substitute the given functions:</p><p>43x - (26,000 + 30x) > 0</p><p>43x - 26,000 - 30x > 0</p><p>13x - 26,000 > 0</p><p><strong>Step 3:</strong> Solve for x:</p><p>13x > 26,000</p><p>x > 26,000/13</p><p>x > 2,000</p><p><strong>Step 4:</strong> Since x must be a whole number (number of cassettes) and x must be strictly greater than 2,000, the company must sell at least <strong>2,001 cassettes</strong> to realize some profit.</p><p><strong>Verification:</strong> At x = 2,001: R = 43(2,001) = 86,043; C = 26,000 + 30(2,001) = 86,030; Profit = 13 > 0 ✓</p><p>∴ Answer: <strong>2,001 cassettes</strong></p>
Correct Answer: 2