Question

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 an equilateral triangle. (Give your answers correct to two decimal places.)

(a) How much wire should be used for the square in order to maximize the total area? m
(b) How much wire should be used for the square in order to minimize the total area?

Answer 1

21.88

39.06

Explanation:

For starters, we say

let the length of side of the square be x

We only that the area of a circle is πr² and we have a circumference of a circle, c = 2πr, so then, we say that

r = c/2π and c = (25 - 4x)

a = x² + π * [(25 - 4x)/2π]², opening the bracket, we have

a = x² + (25 - 4x)²/4π

This function then has a minimum at

x = 25/(4 + π)

x = 25 / 7.142

x = 3.5

substituting for x = 3.5, we have

a = 3.5² + (25 - 4*3.5)²/(4*3.142)

a = 12.25 + (25 - 14)²/12.568

a = 12.25 + 11²/12.568

a = 12.25 + 121/12.568

a = 12.25 + 9.628

a = 21.88

The area is 21.88

At x = 0 the area is just the circle

a = π * (25/2π)²

a = 625/4π

a = 625/12.568

a = 49.73

At x = 25/4 the area is just the square

a = 625/16 = 39.06