1. ## Pieces of Wire

A wire 10 meters long is to be cut into two pieces. One piece will be shaped as an equilateral triangle, and the other piece will be shaped as a circle.

(a) Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the equilateral triangle.

(b) Find the domain of A.

2. Originally Posted by blueridge
A wire 10 meters long is to be cut into two pieces. One piece will be shaped as an equilateral triangle, and the other piece will be shaped as a circle.

(a) Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the equilateral triangle.

(b) Find the domain of A.
Hello,

let x be the side of the equilateral triangle
let y be the perimeter of the circle.

Then you got:

$3x + y = 10 \Longrightarrow y = 10-3x$

The perimeter of the circle is:
$p = 2 \pi r$ That means: $2 \pi r = 10 - 3x \Longrightarrow r = \frac{10-3x}{2 \pi}$

Therefore the area of the circle is: $A_{circle} = \pi r^2 = \pi \left(\frac{10-3x}{2 \pi} \right)^2 = \frac{(10-3x)^2}{4 \pi}$

The area of an equilateral triangle with side x is:

$A_{\Delta} = \frac{1}{4} x^2 \cdot \sqrt{3}$

Therefore the total area becomes:

$A(x) = \frac{1}{4} x^2 \cdot \sqrt{3} + \frac{(10-3x)^2}{4 \pi}$

(b) Considering the values for x you can see that $0 \leq x \leq \frac{10}{3}$