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Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length $ L $ and width $ W $. [Hint: Express the area as a function of an angle $ \theta $.]

$A\left(\frac{\pi}{4}\right)=\frac{1}{2}(L+w)^{2}$

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We're asked to find the maximum mom area of a rectangle that can be circumscribed about a given rectangle with length Big L and with big W trying to work so mhm. So I'll draw a figure to sort of help us explain this problem. So we have our interior rectangle with length Miguel and with big W. And then we want to draw another rectangle around this. It's circumscribed. So it's going to touch this other rectangle. Next comes at its corners like this Diversity. These are supposed to be right angles. Why did have it You like this? Sort of hard to see, but read. Yeah. Now I'll just call the, um this angle here, feta. I'll call this length a this length be this length c and this length D Yeah. Mhm. Now, looking at our figure, it's clear at the sine of theta. Yes, on the one hand, Yes. Why? It's so good. Ah, this is light on is a over big w Do I have to sleep because this angle here is also theater? Yes. Do you ever get woken up in some You additional kids home purse out. Mhm. Mm. Uh huh. Yeah. We also have that the co sign of data mhm is equal to be over Big l Book. Is this our shirt? So solving for A and B A equals big w sine theta B equals big l co signed data and therefore we have the length of this side of the rectangle. A plus B is big w signed data plus big l Co sign data. Yes. Now using the same figure. We also have that the co sin of theta Jordan's is also equal to see over Big W and at the sign of data is equal to D over Miguel. Mm. And you can solve for C and D So C equals big w co signed data and d equals big l signed feta and a symmetry. We had the length of this other side. So the width of the larger rectangle C plus D is w co signed data plus l sine of theta Mhm. Yeah, yeah. Therefore, we have that the area of our outer rectangle A. This is a little a plus B times little C plus d, which you found. This is W signed data plus l co signed data. The election. I need to say just lazing about. Yeah. W co signed data. Plus l signed data is being racist and so we can obtain their area as a function of data. Now, if you simplify, we have a of theta is w squared. Signed data co signed data. Mm. Dropping back, plus W l sine squared data get plus W l co sign square data with this guy. They're like dumb hicks at Louisville. What? Plus elsewhere signed data co sign of theater better. And this simplifies to we can use trigger metric identity is to get a of Fatah equals one half l squared, plus w squared sign of two theta plus W l. Yeah. This is, uh I drove it. Now, just by looking at the equation, we don't even really have to use any calculus. It's clear that, like everybody else, A has a maximum when the function sign of two theta has a maximum. Yeah, of course. The max value of sign of tooth data is one. So I want to find one sign of tooth. Data equals one. This means that to tha tha is equal to pi over two or data equals pi over four. Mr. Yes. Therefore, our maximum area MM is a of pi over four, which is one half times w squared, plus l squared times one plus W l, which is the same as one half times w plus l squared. And so the maximum area is one half W plus l squared about piss inside it.