T-Shirt Profit and a Quadratic Inequality
A student club sells printed T-shirts. Its profit model is P(x) = −3x² + 180x − 1200, where x is the selling price in dollars and P is profit in dollars for a fixed batch. The club wants to know for which prices it breaks even or makes a profit, and whether a price of $35 is acceptable. The advisor warns that the model only makes sense for 0 ≤ x ≤ 50 and that the club cannot set a price that yields negative profit for long. At a meeting, one member claims that because the coefficient of x² is negative, the profit is negative for every price; another claims the maximum profit occurs at x = 60. The club needs a clear inequality solution, the vertex price, and a check of the $35 price.
1. Solve P(x) ≥ 0 and interpret the solution in the context of the price model.
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Start with −3x² + 180x − 1200 ≥ 0. Divide both sides by −3 and reverse the inequality: x² − 60x + 400 ≤ 0. Factor: (x − 20)(x − 40) ≤ 0. The product is nonpositive between the roots, so 20 ≤ x ≤ 40. The club breaks even at $20 and $40 and makes a profit for prices strictly between them. Within the model's stated restriction 0 ≤ x ≤ 50, the valid price interval is from $20 to $40 inclusive.
2. Find the vertex of the profit graph and evaluate the two claims about negative profit and the maximum at x = 60.
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For P(x) = −3x² + 180x − 1200, the vertex x-coordinate is x = −b/(2a) = −180/(2·−3) = 30. Then P(30) = −3(30²) + 180(30) − 1200 = −2700 + 5400 − 1200 = 1500. So the maximum profit is $1500 at a price of $30. The first claim is false because the downward parabola is positive between its roots 20 and 40. The second claim is also false because the vertex is at x = 30, not x = 60; at x = 60 the profit is −1200.
3. Check whether $35 is an acceptable price by evaluating P(35) and comparing it with the inequality solution.
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P(35) = −3(35²) + 180(35) − 1200 = −3(1225) + 6300 − 1200 = −3675 + 6300 − 1200 = 1425. Since 1425 > 0 and 20 ≤ 35 ≤ 40, $35 is acceptable and yields a $1425 profit. This also shows the negative-profit claim fails at a concrete price inside the feasible interval.