Profit In Exercise 38 of Section $9.3,$ we saw that the profit (in thousands of dollars) that Aunt Mildred's Meta lworks earns from producing $x$ tons of steel and $y$ tons of aluminum can be approximated by

$P(x, y)=36 x y-x^{3}-8 y^{3}$

Find the average profit if the amount of steel produced varies from 0 to 8 tons, and the amount of aluminum produced varies from 0 to 4 tons. (Hint: Refer to Exercises $61-64 . )$

32 thousand dollars

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Johns Hopkins University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

so well, the question today is to find their average profits. So we're giving the products Bisset product way. Air X is still product from zero toe eight. And why is our medium produced from 0 to 4? So we have, oh, function P or for Mason? Why in $1000 which is good toe? That's six eggs times y minus, X cubed minus eight. White cube. So we need to find a perfect borough feet off this product. So remember that the four meter for average parole Fitch or average burial it hurts. You say that a P for us is average product. Our every product would be cool to remember that the formula is one over area. This is the area bounded by this. It's bound because we heard about the former examining for right on doing in check grew, huh? This is a bone. You see. Our are means it's his bones off every franks. Why we can write the P. Biggs. Why then the eggs the way. So we have p perfect ry. This is here we would like to find are ours giving us their age and therefore bad. What about a This is the area of Valerie's. Uh, it's sketch finalists, kid for this brown family's area. Okay, am used different carers. Fight yourselves. Kids here. But her eggs like we're 1234 upto aid. This is a eggs, don't we don't have a why, Up to four one tool chri here. Remember that or those variations has to be cool. I mean, the units must be the same spin. So we have some equal variation. So something like this. And then what? I brought eggs. Exes from 0 to 8. That is it from here. Hey, crime. How x here is from here. This is dot newest herb to here on order boat wise from zero up to four. So I mean, is that we need D. C. This area. This is a a our a Good So means that any of this figure this one is zero to age period from Is that is eight times for me. Is that this is goto planted tomb. You wanna talk for area? That's two. Here are 32. The two unit off area. So so for here will get so dis impressed that a p our eight p remember that AP is average pro that is Goto One for that. Too close our A Here we found our AI is felt it too. Then, James, into are rumors for early. This is prone to our brown before x zero. I'm boning for wise zero and used our bone rollerball before wise zero Opera boned is eight and there are burned for exists good to zero and Barbara Bone Okay. And study with X. So this is it for why this is aged for eggs. Then we heard P ex wives goto 36 16 x times away. They didn't mind us exist. Cube then minus 80. Why cube? This is not good. Minus y que Then the extent the way them used for the X here, the exit, then the away good. So we can integrate this world with respect to X. Then after we're integrate with respectful right, this is good too. Won over 32. Integral from 0 to 4. Integral off 36. Thanks. Time was one. Is that that six. Why is constant means that his excess way over to here, You know that is 18 excess square. Why? Yeah, we did 18. I got six over to is 80 eggs is good exist clear and times away, Dale. Minus X cube into your You remember that we add on exponents one and divide. That number means that is extra power for extra poor food. Over. They remind us the Group of eight White Cube. This is constant. Meant that we have 80 eight. Why, Cube? 10 times the kicks in this case. Overuse. We have to have to put the boundaries off eggs, which is good to zero. And the eight floor brawl is zero. A proponent. Easy eight Dan D Y. Because we are triple A rating with respect to away. So Alexis Tay Prisca too won over 32 Integral from 0 to 4. Ah itself. This for you for ex goto age. Here we will have 18 titles. 80 square here is there 64. 80 square is the 64. So we have 18 times is 64. Then why so minus? We heard 844 page to poor fool. Okay. Yeah. Whom then? Here is eight Time of age is 64 Just minus. You haven't minus here. Sorry. Tim's minus 64 64. Why? Cube? Dan a D y we go here is aged world from Yeah, this is good too. He ain't gonna write this one. We have 102 over won over 32. Much 100 partner over. That's too. We can't even trip Great this one with respect toward before carpet this so that I can carry great local Christian out the damn So this you remember This is why Why we are going to integrate with respect to why this is constant, Miss that is White Square over two, which is good toe white scree over too 18 Overtook His name would cover 90 huge time z 64 This is Time Z is no t x times it 64 White square Remember that he I put see or hear 000 That's why there's no minus next time's they always see right? This is zero. This is the only sister That's what So here would have 82 bull for over four. This is a mind us aged four over four. They're insane for off this one because it's constant. It's got away. Didn't mind us 64 over food because you have a wife for for for 64 over four this is 16 16. Why two? There have to pour the very before I she's goto for and zero My bone is zero ever born is for This is good too One over 32 What? That's too. If we put for here is full, it's created Is this 16? So this is nine times 62 not 60 to 64. This is times 64 Sorry it wasn't 64 times 16 for spare is 16 16 minus. This is 44 times four foods for this is 8 to 4. Four So mine us again 16 four times four 4244 So then this is going to be used this way and you will disperse and find the final for you So our average product in 1000 Doris is good too. AP average product This is good too. So here we can divide it. Pride tested through here is that we heard nine times 64 time 16 set Here we can have 64 over Touch to is just two. And here we have 64 times 64 game there you can divide by 20 We have two times 64 and he had this same. We can justice preferred by 1 12 is this is good too Fine, Ari. You get nine times two. This is 18 times 16. There are 16 mine us. This is 64 times 64 128 eight. They remind us here with her over through for cube 16 times from here with her hundreds 28 times before. If we divide touch with Todd to hear, we have 16 times two is two attitude. Then we have to times the frog Cube, which to go to this. And this is good to hear you sing your people Carcaterra this goto 32 towers in loneliness. This system was and Doris, that's 2000 doors starts to spin tales. And Doris can have good too. 32 towers in Doris. Just that 2000 Doris. So this is the end of question Is that average pro index is good toe tarty to Charles in