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Problem 67 Easy Difficulty

Average Production A production function is given by
$$P(x, y)=500 x^{0.2} y^{0.8}$$
where $x$ is the number of units of labor and $y$ is the number of units of capital. Find the average production level if $x$ varies from 10 to 50 and $y$ from 20 to $40 .$ (Hint: Refer to Exercises $61-64 . )$


$14,752$ units


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Video Transcript

{'transcript': "Where are you? We have the program boat average production. You're excusing the product function. Our company have a product, A product function? Yeah, p x appeal for X and Y That is their function off pigs. And why, where x or I are variable. So the company have this production to hundreds times extra points to a point. If I, um times a y two poor zero, this is 0.75. So where the range off eggs is 10 into a 20 in there. Friend off white is good too. 5200. So how to find definite production? We can use this same Milito says the same formula for average average. Very trendy average Very. How do you find average? Very escape with right here. Average production for us is that is average value story. The formula is good. Toe one over. Didn't this warning Very WeII one defy the one dividing by a This a is area times the integral step warranty Go because we have two very ables. There's our function and I would hurt her product function X Why this camera deal from eggs deal for it. So we have our product function what we were looking for. Now we can find a because you have R R. This is Baron Lower Bone, for example, upon for extra job only, For why, and probably for what. So in this case, I want to say area. So this area is area. Oh, just found from these Browner's, you know. Okay, let's try to sketch this bounder. The hair of them use different girls. That can be Thank you. This is not a nice again. Sorry, sir. My very words to using this broke. I can't. There's no program. This is why you see why Access? Sorry. This is a Y axis who had access and the X axis. So let's take that unit is turn. You'll have 10 to 20 you can use. Discourage you. This is 10. Good 10. And put 20 here. Brain onder here. We're gonna hair of 50 0.5200 have maybe start with 50 here and use the same unit 60 70 up. 200 here 5200. So it means that our function x is definitive room pain. Is he finally hear from 10. What? When? On why is the friend of the form of 50 toe hundreds. So it means that area we're looking for is here. This is a So this shows that our A is good too. Our area is good too. You see that if this one is 20 see this side is 20 minus 10 which is Goto 10. Decide is 100 minus 50 which is go to 50. So our area is 50 time Z then. So, friend Mary, we can hurt her be No. Our average is very You average predict a bridge project which is go to one of a P one off a is one of 100 500. Time was the integral from 5200 and Orson cheaper from Kenny told Duane or our p x x wise goto 100 hundreds, Exit four. We can't have any of those. 144 fraction is better. Okay, then times y to Bora, buddy over for it's the vengeful Could be over for 7.753 over four, then indeed, except the except nd What do you want? Good. So this is good toe Come Toto 100 over 500 can pull because this is constantly can bring the constant here before at this sign. If we bring 200 before the integral sign with kids 200 over favorite 100 which is go toe to over five. Then we're gonna see brief by the accretion and get this end table or 50 toe hundreds. We can integrate this first with respect toe eggs. So where we can and when Andy's and then this is control eggs over from if we are OK. This you Do you remember? This is an exponent. Is that table for X? Because why's the questions off X? You can have one refer. It's two for one of the four. Press on over one before plus one. So means that is the five over phone. So we can have one over five or four. Exit to bore favor for Member, that is one plus one over forward. Too slow to fight over four. And then why? To put three over four. Mad here would hurt very ex ago, too. Ex goto 20 on exclude Tim. Then do you want this is gonna be cool too, because this is all we can have. A five. No, we can't have two over fries. Yes, it's a general 50 400 off if we put their funny affects here. We heard twinning toe five over food, but here, this is good toe for over. This is one of a 54 before times five is full times, one over for four or five. We can event. Write it here times for over five, which is Goto eight over. Over 25 not eight. But fear over four or five over five. This is my probation for over five. Here we have twinning to bore four of a fight to par 54 my nurse tending to pour five over whom that we can put the party off weeks. Here are your fix. Been times why bore three over four. The ex so means that we still have ordinary one of very able. Why it's not G expresses the way Thes de Wei. So this is constant. We can calculate it. We can use the Carrick. It has better so that we can be quicker as we can. So this is 80 over 25 10 times twin tow bar fight of ah, full minus 10 245 Oh, full. Then we can integrate. Why? To pull a three performer with respect away. So remember that this is exponent meant that we can add the one here so that this is Goto. Why equal to 73 over four Paswan It's for press forward for seven or full Been over seven over four Then we can put there for you off seven of Verify Goto hundreds And this is Goto 15 which is good to have not enough space C area areas This one them then our average production is good too. O our average production A p This is good too. Uh, 80 overalls wearing face wearing toe for fries over for minus stand for Why? Then time See, We can put their very four right here. This is 100 calories timeto poor seven over four Don't mind us 54 7 over four. This is a pepper bone off White or Bonnefoy 15 bore. Say, Vinnie, over four. Then we have saving or refer. I mean, that we can manage price by four of us for varsity, for sorry. Yes, for over saving. Yeah, So we can use a calculator. This is school started to pour over 25 times seven foot decent character. Authentic. Our creator And then this is going to music They heard with her and the simplification. We can get one. Mr Three, this is Deion 100 the four Terrorism and five hundreds of us three. And if you simplify this one, you're gonna get sited 1000 34 terms in Beatty 133 Dan 23. 23 point Street. This is our average production. This is our average production for this company."}