A large corporation with monopolistic control in the...

A large corporation with monopolistic control in the marketplace has its average daily costs, in dollars, given by $overline{C} = frac{1500}{x} + 100x + x^2$. The daily demand for $x$ units of its product is given by $p = 60,000 - 50x$ dollars. Find the quantity that gives maximum profit. $x = $ units Find the maximum profit. $ What selling price should the corporation set for its product? $

Transcript

00:02 Hi there.

00:03 So for this problem, we're told that a large corporation with monopolistic control in the market has this average daily cost.

00:10 So that average daily cost is equal to 600 divided by x, this plus 100 times x plus x squared.

00:21 Now, once we have this, we are told that the daily demand for this, the daily demand for this is equal to 180 ,000.

00:30 This minus 50 times x.

00:33 So we are asked to find the quality that gives the maximum profit.

00:38 Now, first of all, remember that the profit is the revenue minus the cost.

00:44 Now we know that the revenue is just a pair between the price.

00:49 Um, yes.

00:51 Um, the price and x and the quantity or the demand, and then this minus the cost.

00:57 But first we need to find the cost.

00:59 Remember that what we are given is the average cost, but the average cost is the cost divided by x.

01:05 So if we want the cost in this case, we just need to multiply the average cost by x.

01:12 So that's what we are going to do.

01:15 So the cost function is then 600.

01:19 This plus 100 times x squared.

01:24 This plus x to the three.

01:27 Okay.

01:27 So that is the average.

01:28 Uh, that is the cost, the total cost.

01:31 So now we substitute that into here.

01:34 So then the profit is equal to p times x.

01:38 So that will be then 180 ,000.

01:42 This times x minus 50 times x squared.

01:49 Then this minus the cost.

01:51 Okay.

01:52 So that will be then this minus 600 minus 100 times x squared minus x to the three.

02:01 So the profit is then 180.

02:06 Well, let's simplify things in there.

02:10 Okay.

02:10 This adds, and then we will have this minus this.

02:13 So that will be minus 150 times x squared minus x to the three minus 600...

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