Question

•You have a wire that is 71 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

3s+c = 71
a = √3/4s^2 + c = √3/4 s^2 -3s + 71
da/ds = √3/2 s - 3

clearly a is minimum when s = 2√3

so, the circumference c is 71-6√3

Answer 2

You can assign x for the radius of the circle, or as one of the equal legs of the right triangle, but either way, the answer above is incorrect and unhelpful. To go from x = equal length triangle leg use this total area formula to graph a parabola, then find the minimum value for x which will be your triangle leg length, then find the perimeter of the triangle first, then circle from there.

1/2x^2 + (71-(2x + sqrt2x)/2pi)^2