A manufacturer in a competitive market with sales of x units per week (Se faces

Question

A manufacturer in a competitive market with sales of x units per week (Se faces a market price p = $510 per unit, and has total cost function C(x) = 8000 + 13x + 5x^2. a) Find MR and MC, and where MR(x) = MC(x). b) Find P and MP, the profit and marginal profit functions. Where does MP(x) = 0? c) On the one graph, plot R(x), C(x) and P(x), and on a graph immediately below with the same x-axis scale, plot MR(x), MC(x) and MP(x). Identify the x-values found in parts a) and b). (It is OK to take 0 lessthanorequalto x lessthanorequalto 100, 0 lessthanorequalto C, R & P lessthanorequalto 45,000.)

Explanation / Answer

C = 8000 + 13x + 5x^2

MC = marginal cost becomes :

MC = d/dx(8000 + 13x + 5x^2)

MC = 13 +10x ---> ANS

Now, market price = 510 per unit

So, total rveneue, R = 510x

And MR = marginal revenue = 510 ---> ANS

Now, for MR = MC....

So, we have :

13 + 10x = 510

10x = 497

x = 49.7 dollars

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P = Revenue - COst

P = 510x - (8000 + 13x + 5x^2)

P = -5x^2 + 497x - 8000 ---> ANS

Now MP, so we derive

MP = -10x + 497

And MP = 0 when
-10x + 497 = 0
10x = 497

So, MP = 0 when x = 49.7

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Now, the graph :

For the graph, the software that i use cannot be zoomed out enough.

You can do this one by just plugging them into the graph.

You will find that R and C meet at x = 49.7 and you will find that profit, P will have a maximum value when x = 49.7

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