Thread: maximize and minimize

A wire 20 m long is cut into two pieces. One piece is bent in the shape of
an equilateral triangle, and the other is bent in the shape of a circle. How
should the wire be cut to maximize the total area enclosed by the shapes?

How should it be cut to minimize the total area?

any help?
i got complicated in the numbers :S
root 3 and pi !!

regards

Originally Posted by Abbas

A wire 20 m long is cut into two pieces. One piece is bent in the shape of
an equilateral triangle, and the other is bent in the shape of a circle. How
should the wire be cut to maximize the total area enclosed by the shapes?

How should it be cut to minimize the total area?

any help?
i got complicated in the numbers :S
root 3 and pi !!

regards

$\displaystyle x$ = perimeter of triangle ... each side $\displaystyle s = \frac{x}{3}$

area of an equilateral triangle ...

$\displaystyle A = \frac{\sqrt{3}}{4}s^2 = \frac{\sqrt{3}}{4}\left(\frac{x}{3}\right)^2$


$\displaystyle 20-x$ = circumference of circle

$\displaystyle 2\pi r = 20-x$

$\displaystyle r = \frac{20-x}{2\pi}$

$\displaystyle A = \pi\left(\frac{20-x}{2\pi}\right)^2$

total area ...

$\displaystyle A = \pi\left(\frac{20-x}{2\pi}\right)^2 + \frac{\sqrt{3}}{4}\left(\frac{x}{3}\right)^2$

find $\displaystyle \frac{dA}{dx}$ and optimize

u'll get one value for x only , so how it will be cut to maximize if the value was local minimum ,?

Originally Posted by Abbas

u'll get one value for x only , so how it will be cut to maximize if the value was local minimum ,?

consider the endpoint extrema.

what the domain will be?
is it from [0,20]?

Originally Posted by Abbas

what the domain will be?
is it from [0,20]?

yes ...

x = 0 , all circle

x = 20 , all triangle