## MoreForJoe 3 years ago minimizing area: A 36-in. 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?

1. Dalvoron

|dw:1323260051758:dw| The sum of the areas is$A(r)=\frac{1}{2}(2\pi)r^2 + (36-2\pi r)^2$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. MoreForJoe

what do i do next?

3. Dalvoron

That should get you a value for r.

4. Dalvoron

Then you work out the length $$2\pi r$$, and the length $$(36 - 2\pi r)$$

5. Dalvoron

Actually, according to my sanity test, that gives you lengths longer than 36, so it can't be right.

6. Dalvoron

Nope, calculation error. Never mind.

7. MoreForJoe

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 -..-

8. MoreForJoe

i still dont know where to begin...

9. Dalvoron

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 $$\left(\frac{1}{4}(36-2\pi r)\right)^2$$ Thus, the sum of the two areas is....?

10. MoreForJoe

πr^2+((1/4)(36-2πr))^2?

11. Dalvoron

Precisely. So $A(r)=\pi r^2 + 81-9\pi r+\frac{1}{4}\pi^2r^2$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 $\frac{dA}{dr}=0$

12. MoreForJoe

where did dA/dr come from?

13. Dalvoron

Have you started calculus yet?

14. MoreForJoe

no...

15. Dalvoron

Good grief. I have no idea how to solve this problem without calculus. You've not seen anything like dy/dx yet?

16. MoreForJoe

no... my class is intermediate algebra

17. Dalvoron

Well my apologies, I don't see how it's possible to solve this with just algebra.

18. MoreForJoe

hmm... can you type how to solve this your way and i can just write it down?

19. Dalvoron

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.

20. MoreForJoe

true...

21. alexxis15

WELL THAT'S EASY

22. MoreForJoe

huh?