Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 2.7 Junior Find the answer

A shipbuilding company has an annual shipbuilding capacity of 20 ships. The output function of building $x$ ships is $R(x) = 3700x + 45x^2 - 10x^3$ (unit: ten thousand yuan), and the cost function is $C(x) = 460x + 5000$ (unit: ten thousand yuan). In economics, the marginal function $Mf(x)$ of a function $f(x)$ is defined as $Mf(x) = f(x+1) - f(x)$ .
(1) Find the profit function $P(x)$ and the marginal profit function $MP(x)$ ; (Hint: Profit = Output - Cost)
(2) How many ships should be built annually to maximize the company's annual profit?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

(1) The profit function is $P(x) = -10x^3 + 45x^2 + 3240x - 5000$ (where $x \in \mathbb{N}^*$ , and $1 \leq x \leq 20$ );
The marginal profit function is $MP(x) = P(x+1) - P(x) = -30x^2 + 60x + 3275$ (where $x \in \mathbb{N}^*$ , and $1 \leq x \leq 19$ )
(2) When $x = 12$ , $P(x)$ reaches its maximum value, so the annual shipbuilding volume should be $\boxed{12}$ ships to maximize the company's annual profit.

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