anonymous
  • anonymous
A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=5-x^2 . What are the dimensions of such a rectangle with the greatest possible area?
OCW Scholar - Single Variable Calculus
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
anonymous
  • anonymous
I can't seem to find the the constraint and the good equation to make the problem work :/ except for x=-y - y
anonymous
  • anonymous
or wait forget that last equation :P

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zepdrix
  • zepdrix
|dw:1355978532328:dw|Let's see if we can get the box drawn accurately.
anonymous
  • anonymous
yeah I have it drawn but im still wondering if its symmetrical :P
zepdrix
  • zepdrix
|dw:1355978702013:dw|Understand how I labeled those lengths?
zepdrix
  • zepdrix
Yes it's symmetrical, both corners have to touch the parabola. And in order for the top to be a straight line across, the points have to touch at the same places across the parabola.
anonymous
  • anonymous
yeah it make sense, but shouldnt it be 1/2 x ? or its just 2 rectangles then?
zepdrix
  • zepdrix
It's one large rectangle, touching one end of the parabola and the other end as well. We're basically choosing a point along the parabola. The coordinates that correspond to that point we call (x,y)|dw:1355978868031:dw|And due to symmetry, it'll touch at the same spot on the other side.
anonymous
  • anonymous
yeah it makes sense, the y should be the same on both side, just the roots will be different but the x is the same but on the negative side xD
zepdrix
  • zepdrix
Yah :) So if we want a function for AREA. We need to establish the LENGTH and WIDTH of this rectangle. The Width is pretty clearly Y. What about the Length? Remember, when dealing with length, we don't care about the negative. We can't have a wooden board that is -2 inches long.
anonymous
  • anonymous
yeah so it would be 2x?. However I'll stop you for a sec, everytime I do an optimization i set myself a constraint but there's none here.. am i wrong?
zepdrix
  • zepdrix
The constraint is the parabola, we are constrained to the shape of the parabola. Otherwise the box could just grow and grow and grow.
anonymous
  • anonymous
so the constraint is already labelled as y=5-x^2! smart :P
zepdrix
  • zepdrix
Ah yes, that's convenient :)
anonymous
  • anonymous
so then, if that is my constraint a= 2x(y) and I can replace y with 5-x^2?
zepdrix
  • zepdrix
\[\huge A=2x\color{orangered}{y}\]Yes very good :)
anonymous
  • anonymous
so then i find my derivative as a'(x) = 6x^2 -10x. I get my 0 as +/- sqrt(10/6), do I need to x2 for my full width?
zepdrix
  • zepdrix
Woops, I think your 10x will turn into 10 when you differentiate it, right?
anonymous
  • anonymous
my answer is still wrong x.x and yes it does , hahah
anonymous
  • anonymous
nvm my answer is 2sqrt(10/6), but I have to multiply by 2 right?
zepdrix
  • zepdrix
\[\large A=10x-2x^3, \qquad A'=10-6x^2\]\[\large x=\pm \sqrt{\frac{5}{3}}\]So you're getting a critical point here, yes? Simplify your fraction :D heh
anonymous
  • anonymous
thanks a lot youre REALLLY helpful, i might pass my final thanks to you haha
zepdrix
  • zepdrix
We only care about the positive square root (since it's a length). So I guess I didn't need to worry about the p/m sign. :)
zepdrix
  • zepdrix
So now what we want to do is ummm
zepdrix
  • zepdrix
Plug our critical point into our constraint, to find a corresponding length for Y.
anonymous
  • anonymous
we multiply by 2 because one x is only half the width, than plug it in y = 5-x^2 and were done :D
zepdrix
  • zepdrix
Bah yer too fast! lol
zepdrix
  • zepdrix
So in this problem, they don't want the MAXIMUM area, they just want the LENGTH and WIDTH of the box. We're already half done. We found x. Or rather as you pointed out, 2x. Now we just need Y.
anonymous
  • anonymous
Which i found as 20/6
anonymous
  • anonymous
:) thanks a lot mate!!!
zepdrix
  • zepdrix
Yay team! :) No problem.
zepdrix
  • zepdrix
But again, simplify that ugly fraction! lol 20/6 = 10/3
anonymous
  • anonymous
still ugly in my world. I don't know how you do it but I can never see through this small number of information im given. Or I barely can hahaha. Thanks!
zepdrix
  • zepdrix
\[\large Length=2\sqrt{\frac{5}{3}}, \qquad Width=\frac{10}{3}\]
zepdrix
  • zepdrix
Yah related rates are tough to get a grip on. They show up in many many different forms. :)

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