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Find the area of the largest trapezoid that can be inscribed in a circle of radius $ l $ and whose base is a diameter of the circle.

Maximum area $=\frac{3 \sqrt{3}}{4}$

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University of Nottingham

three of the largest trapezoid that can be inscribed in a circle of radius Al with one base on the diameter. All right, so first thing, let's put this circle somewhere up. Pretend like let's put it in the X y coordinate system. So it's center 00 So right off the bat, we know that it's equation is X squared, plus y squared equals l squared, just in case we need that. Okay, now, once the trap is allowed to have one base on the diameter, so here will be one base, and then we'll just put like this. All right, so we're trying Thio Largest maximize. Okay, Um, largest means maximized the area in this case area of a trapezoid. So the area of a trapezoid is one half the big base, plus the small base times the height. Okay, so the big base is the whole, um, diameter of the circle. So that's equal Thio L Because the radius is l. So that's a constant. So we got one half to l plus little b is from here Thio here. So let's just call this point X Y So little b is two X and then the height is from here to here. Which is why I'm gonna multiply that one half through. So it turned into l plus X times Y. So now what you do is you take the derivative and you set it equal to zero. But we have too many variables. Oh, but we know this so we can use this to get rid of one of them. That's our constraints. So let me write it there. In this problem, X squared plus y squared equals elsewhere. So it doesn't matter which one you solve it for extra. Why? They're both the same. So I'm just gonna solve for why? Set ups, Same amount of difficulty to solve. Same amount of difficulty to plug in is what I meant. Not there the same thing. All right, so the area is l plus X times l squared minus X squared to the one half. Oops. I took the square right there. All right, so I'm gonna take the derivative. Here we go. It's a product, so I'm gonna use the product rule first times the derivative of the second plus the second times, the derivative of the first. Okay, remember, l is a constant just like if it was five or seven. If you don't like the l there, put a number and then switching back to El when you get done. Okay. Where was I? First drew the second, plus the second times. The derivative of the first, which is just one. Alright, this too. And that too will cancel. I mean, rewrite it as a fraction because it's easier to simplify that way. So I get minus x times l plus X over the square root of l squared minus x squared plus the square root of l squared minus X squared over one. I'm going to set that equal to zero. Gonna move one of them over. So I moved it over. It was minus. They were both minus. I made them both. Plus, Now I'm gonna cross, multiply and multiply this side out. I get exhale plus X squared equals l squared minus X squared. Alright, I haven't x squared, so I need to make, but I need to put everything on the same side because it's a quadratic equation. So two X squared plus x l minus. L squared equals zero. So I'm gonna factor that. Let's see, I need a two accent and ex. I need an l and an L. And I need, um, plus and a minus. How's that look? So they'll give me two x square plus two x l minus x l minus elsewhere. Yea, so two x equals l X equals l over to or X equals minus l Okay, that can happen because this is a real life thing, So X equals l over to. All right. So, saying the whole, uh, whole base okay, Bigby remember was to l Little Bee is two X, which is two times l over two, which is one l. And then why equals the square root of l squared minus x squared, which will be the square root of l squared minus l over two squared, which will be the square root of l squared minus l squared over four, which will be the square root of three. L squared over four, which will be the squared of three l over to. So they're the dimensions. Is that what it asked me for? Uh, find the area? Oh, okay, so the area is one half big B plus little b times h So square to 3/4 l times. Three l So three squares of 3/4 l Square is the area

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