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minimizing area: A 36in. piece of wire is cut into two pieces. One Piece is used to form a circle while the other is used to form a square. How should the wire be cut so that the sum of the areas is minimal? What is the minimum value?
 2 years ago
 2 years ago
minimizing area: A 36in. piece of wire is cut into two pieces. One Piece is used to form a circle while the other is used to form a square. How should the wire be cut so that the sum of the areas is minimal? What is the minimum value?
 2 years ago
 2 years ago

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DalvoronBest ResponseYou've already chosen the best response.2
dw:1323260051758:dw The sum of the areas is\[A(r)=\frac{1}{2}(2\pi)r^2 + (362\pi r)^2\]You want this to be a \(\textit{minimum}\). This will happen when the first derivative of A = 0. (though this could also be a maximum. To figure out which, you'd need the second derivative I think).
 2 years ago

DalvoronBest ResponseYou've already chosen the best response.2
That should get you a value for r.
 2 years ago

DalvoronBest ResponseYou've already chosen the best response.2
Then you work out the length \(2\pi r\), and the length \((36  2\pi r)\)
 2 years ago

DalvoronBest ResponseYou've already chosen the best response.2
Actually, according to my sanity test, that gives you lengths longer than 36, so it can't be right.
 2 years ago

DalvoronBest ResponseYou've already chosen the best response.2
Nope, calculation error. Never mind.
 2 years ago

MoreForJoeBest ResponseYou've already chosen the best response.0
i never did one of these problems and im sure he's going to put it on the test so i have to know how to do it ..
 2 years ago

MoreForJoeBest ResponseYou've already chosen the best response.0
i still dont know where to begin...
 2 years ago

DalvoronBest ResponseYou've already chosen the best response.2
The total length of the wire is 36 in, right? That length will then turn into two lengths when you cut it in half. One length will become the circumference of your circle (\(2\pi r)\), and the other will become the perimeter of your square (36  \(2\pi r\)). So now you have: dw:1323260762343:dw I actually made a mistake earlier, I forgot the perimeter of the square and the length of its side are not the same. So the area of the circle is \(\pi r^2\), and the area of the square is the side length squared. The side length is one quarter of the perimeter, so the area of the square is \(\left(\frac{1}{4}(362\pi r)\right)^2\) Thus, the sum of the two areas is....?
 2 years ago

MoreForJoeBest ResponseYou've already chosen the best response.0
πr^2+((1/4)(362πr))^2?
 2 years ago

DalvoronBest ResponseYou've already chosen the best response.2
Precisely. So \[A(r)=\pi r^2 + 819\pi r+\frac{1}{4}\pi^2r^2\]when you simplify, and expand that bracket. What we have is a function, which defines the area in terms of the radius. To give you an idea of what that looks like, go to http://www.wolframalpha.com/input/?i=%5Cpi+r^2%2B%289%5Cfrac{%5Cpi}{2}r%29^2 We're looking for that lowest point on the graph. Where is that point? Well it's at the point where the slope of the graph is zero. The slope of the graph is given by \[\frac{dA}{dr}\], and we want it to be equal to zero, so we say \[\frac{dA}{dr}=0\]
 2 years ago

MoreForJoeBest ResponseYou've already chosen the best response.0
where did dA/dr come from?
 2 years ago

DalvoronBest ResponseYou've already chosen the best response.2
Have you started calculus yet?
 2 years ago

DalvoronBest ResponseYou've already chosen the best response.2
Good grief. I have no idea how to solve this problem without calculus. You've not seen anything like dy/dx yet?
 2 years ago

MoreForJoeBest ResponseYou've already chosen the best response.0
no... my class is intermediate algebra
 2 years ago

DalvoronBest ResponseYou've already chosen the best response.2
Well my apologies, I don't see how it's possible to solve this with just algebra.
 2 years ago

MoreForJoeBest ResponseYou've already chosen the best response.0
hmm... can you type how to solve this your way and i can just write it down?
 2 years ago

DalvoronBest ResponseYou've already chosen the best response.2
It's pretty advanced mathematics compared to what you're currently doing, there's just no way you'd be expected to do it this way.
 2 years ago
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