A stall sells a gadget that costs it 12 dollars per unit. At a price of p dollars it sells 300 − 5p units, for 12 ≤ p ≤ 60. What is the largest possible profit, in dollars? Give the answer as an integer.

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Pricing a gadget

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A stall sells a gadget that costs it $12$ dollars per unit. At a price of $p$ dollars it sells $300 - 5 p$ units, for $12 \leq p \leq 60$ . What is the largest possible profit, in dollars? Give the answer as an integer.

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Hint

Profit is the margin per unit times the units sold; it is a quadratic in $p$ .

Solution

The profit is $P (p) = (p - 12) (300 - 5 p)$ .
Expanding, $P (p) = - 5 p^{2} + 360 p - 3600$ , so $P^{'} (p) = - 10 p + 360$ .
The derivative is zero at $p = 36$ , which is inside the range, and the parabola opens downward.
At $p = 36$ the stall sells $300 - 180 = 120$ units at a margin of $24$ , so the profit is $24 \times 120 = 2880$ .

Answer: $2880$

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