Question

You have a wire that is 100 cm long. You wish to cut it into two pieces. One piece will be bent into the shape of a right triangle with legs of equal length. The other piece will be bent into the shape of a circle. Let A represent the total area enclosed by the triangle and the circle. What is the circumference of the circle when A is a minimum?

Answer 1

let each of the equal sides of the triangle be x
let the radius of the circle be r

area = (1/2)(x)(x) + πr^2
= x^2/2 + πr^2

we know that if the hypotenuse of the triangle is is h
x^2 + x^2 = h^2
2x^2 = h^2
h = x√2

then 2x + √2 x + 2πr = 100
r = (100 - 2x - √2x)/(2π)

then area
= x^2 /2 + π(100 - 2x - √2x)^2 /(4π^2)

simplify, differentiate, set the derivative equal to zero and solve for x
Then find r and the circumference of the circle

Answer 2

Ac = c^2 / 4 π

triangle perimeter = Pt = 100 - c

At = [Pt / (2 + √2)]^2 / 2
... = (Pt)^2 / (12 + 8√2)
... = (1E4 - 200 c + c^2) / (12 + 8√2)

A = (c^2 / 4π) +
... [(1E4 - 200 c + c^2) / (12 + 8√2)]

the minimum A lies on the axis of symmetry ... c = -b / 2a

c = [50 / (3 + 2√2)] / {(1 / 4π) + [1 / (12 + 8√2)]}