Video Transcript

either. In this problem, we have two different plants creating a product and also a Q one is the quantity developed by the very first plan. You two by the second. Therefore we're walk to try to figure out the ideal values if girlfriend want and also que tu come maximize profits. For this reason we should start writing a formula for profit. So profit will certainly be this resources P. Here, let's start by remembering the profit equates to revenue minus costs and also further much more revenue amounts to This is a small P amounts to price, times amount that you're selling. Therefore if us just carefully a substitute, we will certainly be it s okay here. Therefore we will certainly get, uh, instead of p right. We have actually an expression because that pee urine is 60 minus 600.4 Uh que Now instead of Q Member que really equals Q one to add que to miss out on the total quantity, we're told. Therefore that's what I'll do here. I'll make this Q one to add cute, too. Okay, that's that. What we simply did. It's simply pee. So us took treatment of this component Now time Q. And also again. Remember, crucial was in reality Q one add to que tu together a 2nd see what occurred here. This first part is our little P price. Second part is own Cue the amount that we're selling. Okay, Um, the right finest just space our revenue. We market it to subtract off the price cost functions. We have two that them, one because that each plant. Therefore what? to subtract turn off both the them. For this reason they get minus 8.5 minus point. 03 an essential one squared, minus 5.2 minus point. 04 Q two squared. And so all of this, it's attracting off cost. So that is our prophet function. Have to clean it up a tiny bit so the we have the right to do something v it. And it's a many algebra. Um, so, among other points to it is in distributing, he want to ask you come do everything would foil every little thing out below combined. A couple, like, terms you through the numbers, so I'm gonna put dot, dot, period Here. This is the calculus part. That's the hard part Here. I assume that your elder bro's okay, yet here girlfriend can inspect your answer versus mind what friend should finish up with, at least maybe in a various order. However what? you should finish up through at the finish of the day is 60 kyu won, plus 60 Que Tu minus 0.7 key one squared minus 0.8 Q. 2 squared on is your appoints Euro Eights Q. One Q two minus 13.7 Again. The just originates from algebra arithmetic ~ above this prophet function. Currently it'll be simpler to take it derivatives, despite that's the just thing. OK, therefore that's our prophet function. The next thing we do, if you want to discover a maximum, is find crucial points. So let's take it partial derivatives. Let's take it P below Q one to the partial derivative of every this with respect to an essential one. Okay, this an initial term will offer us a derivative of 60 following term zero, however there's no key wanting it the next one. After that, we'll gain minus 0.14. Kyu won next term zero reason there's no key one, and also finally, the second to critical term Que Tu is a constant, for this reason we'll gain minus 0.8 Q two and likewise, let's take the partial derivative v respect come cute, too, and see what we acquire the very first time the will give us miscellaneous is the 2nd 1 below will provide us 60. Currently this fourth term will obtain negative. 0.16. Cute, too. And also then finally, from this term, kyu won this constant. Currently some obtaining minus 0.8 Cute, too. That's our cue one. It's important. It's a cue one. Therefore we have a partial derivatives. Now the whole point of doing this. It's a collection that same to zero in and solve. Therefore we set them equal to zero. And what carry out you execute this by hand? Or the graphing calculator? Several ways to do this theatre is the system. Their equipment of equations. We have two equations in two variables. Variables. Air. Simply you want it? You two watch the stimulate here. Once I to be doing this prior to I accidentally switched them. They've got the not correct answer the an initial time. So make certain you placed them in order once you fix right. However after girlfriend rearrange things, let's see. Us should acquire you all best out this much, so girlfriend should obtain something tantamount to 0.14 key one plus point. Oh, a cute too cool. 60. And over here, 600.8 Q one to add 0.16 q two additionally equals 60. And also if we experienced fury multiply v is just algebra. Um, and the two answers we should acquire are Q one equates to 300 and also cute, too, equals 225. Those space the amounts we should get now, prior to we complete the problem and also say, We're done, us should inspect to make sure that we have uncovered a maximum and also not a minimum. Every right, so these air to visit of answers ideal here, however all we recognize is that that's a vital point. Us don't know for sure that it's a max, so check appear. We'll perform our work check. We'll need this 2nd partials. So looking at p Q. One right below the derivative the that v respect to Kyu Wan. In various other words, ns Q. One Q r. We have the right to see in ~ a glance will gain negative. 0.14 Likewise, Peak, You too cute to let's look under here. The an extremely bottom, the derivative of optimal you to with respect too cute come is right here. We're gonna kids minus 0.16 and now the blended partial P below Q one Q come weaken. Look at p Q one ideal here. Take its derivative through respect to cute too. And also on Lee, the critical term will offer us something it will provide us minus 0.8 Okay, so we usage all this information for the second derivative test, us calculate our capital D, i beg your pardon we deserve to see the an adverse 0.14 negative 0.16 minus an unfavorable 0.8 squared. Um, okay. And we really don't care around the price is lot to be care about its sign. Therefore if we go ahead and also notify this out once you calculation all this out, you gain a decimal. But more importantly, your answer will become less 보다 zero. He'll acquire a an unfavorable answer out of every this, and that's the essential thing. Um, in reality here. Therefore if you run the numbers for a second, we get the best here. And actually D, isn't it? girlfriend should end up with something higher than zero. Again. The number is no as vital as the last science. A D is higher than zero. Let's it is in clear about that. For this reason Deion's of higher than zero. And also it's also noticed the peaky one Q one below was less than zero. And those two bits of info, the 2nd derivative check tells us are sufficient to conclude the this critical point right here is important a neighborhood Max.


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So we have discovered a best key. One should be 300 friend two must be 225 and