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Problem 42 Hard Difficulty

A rectangle $ \Re $ with sides $ a $ and $ b $ is divided into two parts $ \Re_1 $ and $ \Re_2 $ by an arc of a parabola that has its vertex at one corner of $ \Re $ and passes through the opposite corner. Find the centroids of both $ \Re_1 $ and $ \Re_2 $.

Answer

$R_{1} :\left(\frac{3 a}{4}, \frac{3 b}{10}\right), \quad R_{2} :\left(\frac{3 a}{8}, \frac{3 b}{5}\right)$

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Video Transcript

in this problem were given to regions are too and are one in direct angle for my total regions. And where has to find a central right off each individual region. Now we're gonna work with region. One are one and the outdoor rectangle and no, from those two we're gonna get information about region are too. Let's start from the first record rectangle. Actual list are from writing de question off this part. But we know that that will be in general for wise sickle to K expert, and we're giving corn. And it's we know that this parable of Mrs uh point a would be if you plug does and we see that beat and should vehicle to k Time's a scribe and therefore must be fine, Kay to be equal to be over a squared. So the question of parabola is Then why's he going to be over ace Grade. Times X scraped. All right, we have this inherent. Now let's find the area off Region one. So we are now forming 10. Sheet off. What? The X and we're something those up until we get the area off the whole region. Basically, Then we're intervening. That's the area. One area Frieda want the lettuce off. Interrogation will be between zero and a since eggs goes from X goes from zero to a and e de function. Here is why Isaac. Okay, excursion. And we know what case. So the area will be then be in charge. Go from shoots a B or a scarred X squared touch D X that will be equal to be over a spread x cubed over three or exchange between Germany and doing this. We find the area off that region to be eight times beat over three. All right, now, let's fine. Um, the area of the second region banality. The area of second region is total area minus area. First regional war. Total area is area of a rectangle with sites I am beat. So it'll be a times B minus a B over three. So it'll be good to a B over three. All right, we know that export will be able to form water the corresponding area integral from A to B. Those are the interrogation limits. X ffx d x and Y barbell. Denby. Oh, for one over area inter go from A to B. What health f skirt of X times the X Now we have everything. Let's just plug devalues ordinance. And, um, we are interested in region one right now. All right, so export for that would be one over corresponding area. So one over a B or three. There will be three over a V inter go from zero to a. Since those are the limits for X, uh, ex terms, this is the function if a fight so x times be over a scribe Times I described the x B over ace card is just a custom. We can take it outside in triangles or three over 80 times be all race card. We have X cube inside integral Saudi entire narrative. For that would be extra fourth over four were exchanges between your own A And when we do the calculation, we see that x four for region Woman is looking at it that three a over four. Now let's look at why bar for a reason? Well, it will be three over a b inter go from zero to a one house off, uh, dysfunction square so that his be spread derided by 84 extra Ford, the ex that will be equal to free Over to a B um, times be scared or h of the fourth extra 50 light Fine for exchange between zero A calculating best be found. What bar for region Want to be located at three B over 10 sudden century off region Willis or center off mess for region. What or one is the tree? A over four and three be over 10. So visually, uh, it is somewhere on here, that is three a over four. And this is three B or 10 and this is sentry off Region one. Now you know that we have a rectangle. The art director. I mean, on a central off a rectangle is, at its symmetry, anxious. So this art director Noah symmetric with inspector this access in horizontal direction and in vertical, it systematically distracted the sexes. So this is the overall central off this object and dick ordinance off that point would be aye or two and B or two. Since the total length and resentful uh, direction is a vertical direction is beat. All right, we know that, um export or roll central eight times the total area isn't culture Central region one times corresponding area plus Sentra off region, too. Times is corresponding area in extraction and saying for a wide direction. How do I get that? Well, this is basically Samos saying that export is equal to and time's, uh, ex do you want it by total M or this is equal to saying Thanks, Ethel. Thanks. Delighted fo x. All right. No, overall, central is the central right off the out correct angle. And it is an ex and extraction. The Centauri is located at a Were too. So we had they were to total area off. Direct angle is a times B. This is a call to Central Region one. We found us to be a 303 or four south three over four area off. Region one is found to be a temp. Your three. Plus we do not know. And we're looking for centuries off region to an area freezing to is to a B over three. Now a bees will cancel out. And from this we found eccentric ride off region too to be three a over eight. And if he applied the same formula, so wipe four times a total. Why barbed aural system as why boar off region one times the area of reach, want and orb are freezing to terms. Area freezing too. We see that, uh, this is why bar or roll region ist be over to area is a times B. Why borrow the first area or region is found to be three V or 10 an area That region is a times B over three plus. Why to bar is what we're looking for in a two is to a B or three babies again Well can slap for miss be found one far region to be at three b or five So here we can see that then center of mess off Region two is located that at three a or eight and three be over five. So the answer is dez for region to and this for region one