Okay, so for problem 31 there's two parts parts A and B apartment. It wanted you to show that doubling both labour and capital results of doubling production. T so labour being X and capital being Why so? Basically, I want you to show that p of two x two y is coming equal to two times p of X y. So let's go ahead and write that know sweet have k two X to the A times to why? To one minus a So this condemn be rewritten as que tires to a X A what? Because the X there the first time. And then that could also be written as times two. Why to the first power times two y two negative a power. That's because when two eyes one minus a power, that's the same thing. Is writing is to watch the first times to watch in the debate there for the next set. What we can do. We can go ahead and move that, too from the two wire front because it doesn't have any exporting when it was going to K Times two today, next today times why, and it is too wide to the negative they can down to go to the denominator since it's the negative exponents and we could just make the expletive positive. Once acquitted, this is gonna be all over. I'm also to make that two to a power. And why today? Parts. I split that up, seeing my house, but of dec's. So once you see that you see that right here, to to the A to today cancel out. And then for this next set, I'm gonna go ahead. Bring that. Why today? Back up. This is gonna be to to decay two times k x a. Do you know why? Tells what's it? Negativity. So there you could say that is two k x a Why? To the one minus a power wants the game, and you could see that checks out. That is equal to two p of X y okay, Part me. They said, uh, kz could want 20 and a zygote of three force So Israel I r p of X alive. So it's now nearly equal to 1 20 x to the three force times alive to the one minus exits one minus de force, Or why to me 14 Okay, So that's a P of X Y and G of X y. They said that they every cost $250 for each unit. So that's 250 x then capital cost for in geology for each units or plus 400 Why? And the costs cannot exceed $100,000. So it only minus 100,000 that must be equal to zero. So no tell of p was read was blamed it on the G. So Delle api is gonna be equal to 120 times three Force X. So exit e force of minus lawyer is gonna be exit negative one force. Then you can keep the why to the one for had direction. Waas 120. How's life worth Institute taking this driven Inspector Warren Now exit the three force. Why is his negative G force that would be equal to Well, I am Joe turns 215. No, I had direction plus lander times 400 in tow. J. Edgar No said the components equal to each other and I'm also gonna do some multiplication. So 1 20 times three of our four is gonna be equal to 90 So it's 90. And then, since it's exits to native 14 I'm just put that indigent on Lady. This is 92 y suing for over extra one poor. So I just switched the science and puts it in a nominee that's gonna be equal to Lander Times 200 and they 1 20 before that's gonna be equal to 30. So, so angry. 30 extra tree force Over. Why? To the three force Aston only equal to planned times for Okay, So what I'm gonna do now, I'm just gonna solve you Cycle, ada. So on the right side, I'm gonna get 90. Divided by 2 50 That's just give me 9/25. Why? 214 over. Excellent For is gonna go to Atlanta. 30 over 400. This is gonna be 3/40. So 3/40 extra to be force who are didn't force. It is possible that you could surrender Sinaga cities equal to each other. So when I wanted one for over 25 exit award for is going to be quits, it three Excellent. Three course turns 40 over 40. Watt said if the force now what? I'm gonna do is I'm going to cross multiply both sides. So we're gonna get 40 times and I that's gonna be 300 3 60 mother. So 3 60 and then wide some wide to the 14 times. Watch it. 3/4 we're gonna add the exponents so it's gonna be wide today. 44 So it was a lie is going to be equal to 25 times Street 75. Thanks. And then what we can do? We can divide that d 62 other side and get that wise and equal to 75/3. 60 x, which is just 5 24 Okay, are so let's we're gonna do is we're going to plug our Y into our G of X likely equation. So to 50 X was 400 times 5 24 That is gonna be equal to $100 so we can try. Simplify that out 400 times five divided parts of before. Well, I'm just gonna leave it over 24. Pronounceable is gonna be 2000 over 24. Plus, it's only to treat me over to you for where? So 6000 x over 24 to meet with 100,000 So 1000 X over 3400 and then we can simplify that Excellent legal to 300. And then why is it going 5/24 x so we can just multiply that by five and divide by 24? Why isn't equal to 62 0.5? So that means for maximum production level, um, and not exceeded cost over 100,000. We need brown. Oh x hours of label x 300 hours of labor labor Sorry in 62.5 on UNESCO capital invested.