Find the minimum area, function, derivative

Question

A piece of string 60 cm long is cut into two, one piece being is used to from an equilateral triangle and the other in the form of circle. What should be the perimeter of the equilateral triangle and the circumference of the circle so that the sum of the areas is minimum? Find the minimum sum of the areas. Express your answers in the form of π.

Answer 1

triangle area = At = bh/2 = s(sqr3)/2/2 = (sqr3)s^2)/4 where s=the length of a side of the trianlge

circle area = Ac = pi(r^2) where r = the radius of the circle

circumference of the circle = 2pi(r) = 60-3s

perimeter of the triangle = 3s

2pi(r) = 60-3s

r = (60-3s)/2pi

Ac = pir^2 = pi[(60-3s)/2pi]^2 = (60-3s)^2/4pi

At + Ac = (sqr3/4)s^2 + (1/4pi)(60^2 -360s + 4s^2)

= (sqr3/4 + (1/pi)s^2 - (90/pi)s + 60^2/4pi

take the derivative and set = 0, solve for s, then multiply by 3

(At +Ac)' = (sqr3/2 + 2/pi)s - 90/pi = 0

s = (90/pi)/(sqr3/2 +2/pi)

3s = (270/pi)/(sqr3/2 +2/pi) = perimeter of the triangle

60-3s = the circumference of the circle = 60 - (270/pi)/(sqr3/2 + 2pi)