Average Profit The profit (in dollars) from selling $x$ units of one product and $y$ units of a second product is

$$P=-(x-100)^{2}-(y-50)^{2}+2000$$

The weekly sales for the first product vary from 100 units to 150 units, and the weekly sales for the second product vary from 40 units to 80 units. Estimate average weekly profit for these two products. (Hint: Refer to Exercises $61-64 .$ )

1866666.678

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Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

Boston College

{'transcript': "All right. So we have these two products and the prophet function. So X is the first product. Why's the second product? Minus X minus one hundred, one hundred square minus Y minus fifteen squared. Plus two thousand. And let's see home. The first product varies from one hundred one fifty and then the second from forty to eighty. Okay, so what is thie area? So we're doing an average value. So we've gotta divide by the total area of this rectangle fifty times forty. That should be two hundred forty times. I'm sorry. Uh, yes, four times five is twenty. And then you two zero, two thousand. So the average value is going to be one over two thousand. The area, and then we got actually do the center girl one hundred one fifty, forty to eighty of this crazy function. No, no, no. So we have a minus minus here. I'm just going to put that minus out here so that this is actually just x minus one hundred squared. Plus, why minus is he squared minus two thousand. And you don't really need to do that. I just don't like the negatives with the variable. So this is D y the ex. Thank you mind. Over two thousand. All right. Anti derivative. With respect to what? So this is just going to be Why? Times X minus one hundred squared. Plus. All right, here we go. There should be one third. Why? Minus fifty. Cute. Minus two thousand. Why? Evaluated from forty eighty? Yeah. Yeah, Jax. Excellence. Plug in forty maybe. For why so the first time I still have the same girl. First time we have just the difference of forty extra minus one hundred squared. And then one third artist, we're gonna die. Aerated eighties. That's going to be thirty. Cute thirty cubed and then negative. Ten. Cute. Just thirty. Cute. Minus negative. Ten. Cubed man. Just two thousand times forty minus two thousand. I'm supporting tensions. Eighty thousand. Okay. And good. One two thousand. Okay, so now we need to take another anti derivative. Well, we could probably evaluate this. It's a thirty. Cute is well, twenty seven with three zero. So I was twenty seven thousand then plus ten cubed is thousand to twenty seven thousand. Plus one thousand twenty eight thousand. Okay, so here we get forty over three X derivative. Still you enter you'LL sign anymore. Okay, so forty over three x minus one hundred cute plus one third. We said this was twenty eight thousand that'LL get X minus eighty thousand that also get next evaluated for one hundred twenty fifty. Okay, this point we could, despite all this in and we get Okay, So this is the dollar amount. Approximate the average profit is nine hundred thirty three dollars and thirty three cents."}