## hersheys06 4 years ago A piece of wire 25 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (a) How much of the wire should go to the square to maximize the total area enclosed by both figures? ANS= 0 m *** (b) How much of the wire should go to the square to minimize the total area enclosed by both figures?

1. satellite73

if you want a guess i would say put everything in to the circle and forget about the square

2. satellite73

3. satellite73

probably all in the square to minimize rigth? just a guess though

4. ktklown

that's my guess too, but, i'll bring mathematica out again ;-)

5. ktklown

yeah, mathematica says to put 100% into the square

6. ktklown

In[940]:= Minimize[{$Pi]r^2 + x^2, 4 x + 2 \[Pi]r == 25}, {r, x}] Out[940]= {1/16 (625 - 100 \[Pi]r + 20 \[Pi]r^2), {r -> 0, x -> 1/4 (25 - 2 \[Pi]r)}} 7. hersheys06 How much of the wire should go to the square to minimize the total area enclosed by both figures? i think it requires a numerical answer 8. ktklown all of it. 25 m 9. hersheys06 it says it's incorrect :S 10. ktklown hmm 11. ktklown oh now i get x=3.5, that's one side of the square 12. hersheys06 sorrry, still incorrect T___T; 13. ktklown well it's 4 times that i just gave you one side 14. hersheys06 ohh, i did the area (3.5)^2 instead of (3.5)*4. thanksss 15. ktklown did it say that was right? 16. hersheys06 yess 17. robtobey (b) How much of the wire should go to the square to minimize the total area enclosed by both figures? The exact answer is:\[\frac{100}{4+\pi } \text{ meters}$or 14.002479 meters to 8 digits.

18. robtobey

Take the derivative of the expression of interest$D\left[\left(\frac{x}{4}\right)^2+\frac{(25-x)^2}{4 \pi },x\right]$set it to zero$-\frac{25-x}{2 \pi }+\frac{x}{8}==0$and solve for x.$x\to \frac{100}{4+\pi }$A plot is attached.

19. robtobey

Using Mathematica's Minimize function:$\text{Minimize}\left[ \left\{\left(\frac{x}{4}\right)^2+\frac{(25-x)^2}{4 \pi }\right\},x\right]\to$$\left\{\frac{2500+\frac{40000}{(4+\pi )^2}+\frac{10000 \pi }{(4+\pi )^2}-\frac{20000}{4+\pi }}{16 \pi },\left\{x\to \frac{100}{4+\pi }\right\}\right\}\text{ //N}$