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pansy2015 · Oct 28, 2011, 04:55 PM

A company manufactures and sells x television sets per month. The monthly cost and price-demand equations are C(x)=72,000 60x p=200-x/30
1. Find maximum revenue
2. Find the maximum profit, the production level that will realize the maxium profit, and the price the company should charge for each television set.
3. If the government decides to tax the company $5 for each set it produces, how many sets should the company manufacture each month to maximize its profit? What is the maximum profit? What should the company charge for each set?

jcaron2 · Oct 28, 2011, 09:56 PM

I'm going to assume you're familiar with calculus. I'm also going to assume that there's supposed to be a + sign between the two terms in your cost equation.

1. The revenue is the price per unit times the total units sold. So in this case, we can write the revenue as

$R(x)=p(x) \cdot x=x$200-\frac{x}{30}$=200x-\frac{x^2}{30}$

If we want to maximize R(x), we find the value of x where its derivative is zero:

$\frac{dR(x)}{dx}=200-\frac{x}{15}=0$

$x=3000$

So maximum revenue is achieved when the company sells 3000 televisions. The value of that revenue is

$R(3000)=3000$200-\frac{3000}{30}$=300000$

2. To calculate profit, we use a very similar procedure. Only the initial equation is different. Total profit is equal to the difference between revenue and cost.

$P(x)=R(x)-C(x)=x$200-\frac{x}{30}$-$72000+60x$$

Again, to find the maximum of this function we simply set its derivative equal to zero.

$\frac{dP(x)}{dx}=0$

I'll leave it up to you to do this one. You can post your answer if you want me to check it for you. Likewise, if you get stuck and are unsure how to proceed, let me know what's tripping you up and I'll be glad to provide more guidance.

3. This is just like part 2, except now the profit formula is slightly modified. There's an additional -5x because they lose $5 to the government for each unit they sell. (In reality, we really are adding an extra +5x to the cost equation, but since cost gets subtracted to calculate profit, it ultimately ends up being a -5x in the profit equation). You simply need to include that extra term in your equation and repeat the same procedure as above.

As for the price the company should charge, I'm not really sure how to answer that. Are we supposed to assume that they'll sell ALL of the televisions they produce, regardless of price? In that case, they should put a price tag of $1 billion each. Or $1 trillion. Or infinity. Obviously that's not very realistic; they wouldn't sell too many TVs at a trillion dollars apiece, so that's probably not what they mean. I'd probably just say they should charge an extra $5 over their previous price (i.e. p=205-x/30). That's just a guess though. Seems like a poorly worded question if you ask me.