Posted by **Dan** on Monday, November 19, 2012 at 12:28pm.

A 100 inch piece of string is cut into 3 pieces. One piece forms a circle, one forms a square and 1 forms an equilateral triangle. If the perimeter of the triangle is equal in length to the perimeter of the square how long should each piece of string be to minimize the total area? What is the diameter of the circle? How long is each side of the square? How long is each side of the triangle?

- calculus -
**Reiny**, Monday, November 19, 2012 at 2:24pm
let each side of the triangle be 4x (you will see why ...)

then each side of the square is 3x (ahh, how clever)

height of the triangle is 2√3x , by Pythagorus

let the radius of the circle be r

2πr + 24x = 100

πr = 50 - 12x

r = (50-12x)/π

total area (T) = πr^2 + 9x^2 + (1/2)(4x)(2√3x)

= π(50-12x)^2/π^2 + 4√3x^2

= (2500 - 1200x + 144x^2)/π + 9x^2 + 4√3x^2

d(T)/dx = -1200/π + 288x/π + 18x + 8√3x

= 0 for a min of T

timex π

-1200 + 288x + 18πx + 8√3π x=0

x(288 + 18π + 8√3π) = 1200

x = 3.092

r = 4.104

so diameter of circle is 8.21

perimeter of rectangle is 12x = 37.106

perimeter of square is 37.106

check:

2πr + 12x + 12x = 100

## Answer This Question

## Related Questions

- Precalc - You have a wire that is 100 cm long. You wish to cut it into two ...
- Precalculus - There is a piece of wire 200 cm long. It needs to be cut into 2 ...
- calculus - A five feet piece of wire is cut into two pieces. One Piece is bent ...
- Calculus - a piece of wire 12 ft. long is cut into two pieces. one piece is made...
- Calculus - A piece of wire 40 m long is cut into two pieces. One piece is bent ...
- Calculus - A piece of wire 40 m long is cut into two pieces. One piece is bent ...
- Precalc - •You have a wire that is 71 cm long. You wish to cut it into two ...
- math - A 24-inch piece of string is cut into two pieces. One piece is used to ...
- math - The perimeter of a square piece of land is twice the perimeter of an ...
- Calculus - A 30-in. piece of string is cut into two pieces. One piece is used to...

More Related Questions