Problem 73 Hard Difficulty

The upper right-hand corner of a piece of paper, $ 12 $ in. by $ 8 $ in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose $ x $ to minimize $ y $?

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Wen Z.

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Amrita B.

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Linda H.

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the upper right hand corner of a piece of paper. 12 inches by eight inches. As in the figure is folded over to the bottom edge. How will you fold it so as to minimize the length of the fold. In other words, how would you choose X. To minimize why? And this is the statement we're going to stick with. This one here is clearer than the other one that is. We want to choose X in order to minimize value of Y. And for that we need to write why as a function of X. And they apply and then apply calculus to minimize the dysfunction. So this is the situation geometrically. And as we can see the fall the a sheet of paper and we found a right for angle here which is the reflection following the lion of length Y. Of the right triangle with the dot lines. So these two right triangles are the same this with the red lines and the other with this red line as his thought lines here because we have a right angle here of course. And that right triangles. Just this one here. So the triangles are the same. We have these length, we are going to go Z. That is the length of the start line here. The other dot line is eggs. And this diagonal red red line is why? Yeah. So we know that here, we have the total length minus x. That is eight minus x. Mhm. If we draw perpendicular lion here from this point here to the other side of the sheet of paper that is. We draw a vertical line. We have then um another triangle here and this. I'm going to call this distance from here to here. That is to this note, we're going to call it in And these other two here from this note here to the age of the paper, we're going to call it M. And from the figure is clear that M plus M is Z. Okay, this is something that we can see clearly clearly in this figure. Okay, so we want to write one in terms of X. And for that we are going to use these right triangles we have thrown here. We have the first one is deals with the red lines. The other one is this here using their bottom right corner of the paper. Mhm. And from the observation that is rectangle here up is the same as this rectangle with red lines. We can see that this side right here is of length X. Because it's the same as this when we fall the paper. Is this just the same? And the other side is this one is this one? That is this is see here, mm haven't done this when we now have all the red triangles. I'm going to draw them here. Mhm. The first one is the the single and uh red triangle. This one here. Yeah, with the right triangle here, the right angle here and X. Z. And y. The other one that's called these one what? Then we have the other is this here. So this is like this, this is the right angle and we have eight minus X. We have X. Here as a hyper luminous. And the other side is m You're going to call it too. And this one here, yes. Just like this with the hype oddness. See here is the right angle and here is we have called it in and we know that this side here is eight because he's the with of the piece of paper and we are going to cold is three. So we have now the three red triangles we're going to use and this statement here. So first yeah We use this one from yeah from right triangle one. We have the first relationship is X square plus Z squared equal y square. Okay remember the hype redness of this triangle here is why that is why the equation is like this by to ground zero. So we have this one. Mm So right I the first one, the second thing we're going to write is this Relationship here from these triangles of from two. Okay, mhm Eight minus X square plus m square. But like square. Yeah. And from the right triangle. Uh this one here the third one. Yeah. And so we have eight square plus and square equals Z square. Yeah. Okay, so we have this and we call this equation here two. Okay And this equation here three roman meditation. So these are three equations from the three right triangles we constructed from the figure. And so we start with now we recall that we want to write wine terms of X. And from the first red triangle we have Y here in terms of X and Z. So we've got to put Z in terms of X. And we will be done Yeah. And we'll be done for the first part of course. And then we have Z to put me in terms of X. We have this third equation you see is written in terms of end but we know em in terms of X from two and we know that N. M. And Z are related but this is rich. That's gonna give us what we want. So now we start with the second equation too. Mhm. Okay so we get from this equation here that eight square minus 16 X plus x square equals plus m square equals X square. Ex queries cancel out. So we get from here that uh m square Plus 64 is equal to 16 eggs. Yeah you have the situation here and we can get a little bit further, we get that X. Is equal to m square plus 64 over 16. Or sorry maybe it's not accept. We want Sorry sam. So we add the other step we can do is M square Equals 16 x -64. And then mm can be written as the square root of 16. My 16 X -64. We take a con factor 16. Here we get x minus four. And so this is four squared of x minus four. And in order to do this, We get to impose the condition that X -4 is positive or zero. So X minus four squared. And then record zero that his eggs Is at least four. That's very important condition. That is we cannot do this. Uh Less than four. If you do this, less than four we will we won't have a geometrically possible figure. Okay, so we have these we have these partial results were going to use after and then we have written mm in terms of X already and these were going to call four. Okay, Now use the other equation here is three. Yeah. From three. We get that. Uh eight that 64 plus N square equals E square. But we know that N plus M equals E. And that is the observation we do right here following the construction and plus M. Give us all this whole land sea here so we can food and as Z minus M and 62. That here. So. And then 64 plus Z minus m square is c square. That is 64 plus C square minus two. Zm plus m square equals the square. See squares cancel out. So we finally get to Z M equal 64. Blossom Square. And that is why we are going to use this preliminary result here. M square plus 64 60 next. So we can as we can see here we have 64 M squared plus 64. So we can replace that with 16 eggs. Yeah That means that two ZM is equal 16 eggs. And for that Z is 16 X over two M. Which is eight X. Over M. And we get to put em. Now we found here mm you've seen four. Z is equal to eight x over for Square. It affects -4. That is Z is eight over four is too. So we get to in the numerator had two eggs over spirit affects one's four. And that's it. So we have see in terms of X and we are done too right? Why? In terms of X. So This is question one. So from one and five, That is this one here we obtain that? The C. X square plus the square. That is X square plus two eggs Over square. It affects -4 square equals Z square. A white square, sir. Mhm. And then why square is X square plus four X square over square. Sorry, over x minus four. Sure that is X squared. Common factor of one. Plus for over x minus four. That is six square times X -4 Plus four. Over X -4. These four canceled here. And so we get X square Time Sex Over X -4. And so why square is equal to X cube over x minus four because all the lens are positive. This we can take root and get y equals square the facts cube over X -4. And for that s get it, get it x gonna be positive and greater than for in fact very left for would be the same as positive. And I put it right here. So this is our equation of Y. In terms of X. And the condition will be excreted for And now for these two have sense, X cannot be four. In fact can be must be creators strictly than four. And from this forget that it is positive and the negative excuses. Makes sense. So this is our equation. Now we got to apply calculus here to find the maximum. Well the minimum value of Y and the corresponding X. So for that we need the derivative. So the function we are going to use this square root of execute Over X -4. And we need to calculate the derivative of this function. Let's write it as X cubed over x -4 to the one half. And with that mhm. Here, I'm going to write that. We're going to minimize f of X Is the goal. So the derivative is 1/2 times X cubed over x minus four to the one half minus one times the derivative Of XQ over X -4. Following the chain rule. So this is 1/2 times XQ over X -4 to the negative 1/2 times deserve a tyvek ocean of two polynomial. So is three X squared times six minus four minus x cubed times one over x minus four square. So this is 1/2 times. This can be written as one over the square root of X cubed over X minus one. Then we can invert the fraction. So we get square root of x minus four over the square root of X cubed Times. In here we have a three x cubed -12 minus 12 mm X x square minus X cubed over X -4 sq. Mhm. And so this is 1/2 times X -4 to the one half times these can be written as X correct effects because excuses exports and sex. And we separate the square root and we get this times two x cubed -12 x square over Eggs -4 Squared. Yeah. And this is one half times X minus forced to the one half over X times X to the one half times uh 22 x square. Common factor of x minus six Over X -4 square. And here we can simplify things here. Okay, so we can say this is the to cancel out This sex too. In fact we can let it as three houses better simplify with this square and this X -4 to the one. How with this one. So we get in moderator X two D two minus two thirds minus three. Third three half. Sorry That is one x. M5 with his ex and ex over X to the one half is excellent one house. So this is squared of X X -6. And in the denominator will have X -4 to the 3/2. That's our derivative. Finally. Yeah. Okay. and is zero when X zero or eggs equals six? It is not possible. Yeah That it's equal zero Because eggs got to be greater than four. Mhm possible at X equals zero because okay then X equals six. It's the only critical point. Mhm. So the candidate to have for the function of to have a minimum to know what kind of critical point this is. We get to find the second derivative if we do that we get that second derivative of F. At any X is equal to 12 over mhm. Square root of x times x -4 to the fifth. And then the second derivative at the critical .6. Give us 12 over Square root of six Times 6 -4 to the 5th which is a positive quantity. So for X equal, see ffx has minimum value. And that's what we're looking forward. There is eggs Equal six. And for that why is a minimum in terms of X. And so uh if x equals six we can find why you seen This relationship here. So why is the square root of execute over X -4? So the corresponding value minimum value of F of y? Yes, the square root of X cube six to over six minus four right here, That is six scores of six over six final fours two squirts of two and 63 times too, so we get six quarts of three times square root of two over scores of two and so six square to three, which is about been 10 point 39 23 inches. So they see the minimum value of why and It is attained when x equals 6".