Raymond B. answered • 03/23/22
Math, microeconomics or criminal justice
height of the triangle =(sqr3)/2)s)
area of the triangle = half height times base = (sqr3)/2](s^2)/2 =(sqr3/4)s^2
for 60 feet, all used for the triangle, each side would be 60/3 = 20 = s
and the area = (sqr3/4)20^2 =400 x about .433 = 173.2 ft^2
exact area = 100sqr3 square feet
you want the minimum area
First guess or inclination is use
virtually all of 60 feet for the triangle and a tiny nearly zero micro inch for the circle if you have to cut it into 2 pieces
maximum area would be when you use all of it for the circle = pi(r^2)
where the circumference = 60 =2pi(r) and r= 30/pi.
area = pi(30/pi)^2 = 900/pi = about 286.5 ft^2
the more of the string used for the triangle seemingly the less the total area.
But use 55 feet for the triangle, then the side = 55/3, and area = (sqr3/2)(55/3)(55/3)/2 = about 145.54
add the circle's area where 5 = 2pi(r), r=5/2pi A=pi(55/2pi)^2 = 25/2pi = about 3.979
total areas = 145.54 +3.98 = 149.52 which is less than using all for the triangle
for a fixed perimeter, equilateral triangles have the smallest area of any regular polygon, while circles have the largest area.
Area of the triangle = (sqr3/4)s^2 where s = one side
Area of the circle = pi(r)^2
r= radius. Circumference of the circle =2pi(r)
2pi(r) + 3s = 60
r = (60-3s)/2pi
Total areas of the circle & triangle = T = (sqr3/4)s^2 + pi[(60-3s)/2pi]^2
take the derivative with respect to s and set = 0, solve for s,
T' = (sqr3/2)s -90/pi + 9s/4pi = =0
s = 180/(sqr3 + 9pi)
3s = string for the triangle = 540/(sqr3 + 9pi)
=about 3(13.51) = 40.53 feet for the triangle
leaving about 19.47 feet for the circle
Total area =triangle area + circle area = [(sqr3)/4]s^2 + pi(r^2)
Total minimum area
= (sqr3/4)[180/[(pi)(sqr3)+9pi^2] + pi[(60-3(180/(pisqr3 + 9pi^2)]^2 = about 88
