the sum of perimeters of an equilateral triangle and a circle is 'k'(constant) Prove that their combined area is - Maths - Application of Derivatives

Question

​the sum of perimeters of an equilateral triangle and a
circle is 'k'(constant) Prove that their combined area is minimum
when the side of the triangle is 2root3 times the radius of the
circle

Aarushi Mishra · Accepted Answer

Givenperimeter of circle+perimeter of equialteral triangle=kperimeter of circle=2πrperimeter of equialteral triangle=3a,         where a is side of equilateral triangle2πr+3a=ka=k-2πr3 _______________(1)Area of circle=πr2Area of equilateral triangle=34a2Sum of area A=34a2+πr2From equation 1A=34k-2πr32+πr2For maxima or minima dAdr=0dAdr=34×2×k-2πr3-2π3+2πr=32πr-k9π+2πrdAdr=32πr-k9π+2πr=0Put value of k it was k=2πr+3a32πr-2πr+3a9π+2πr=03-3a9π+2πr=0-a3π+2πr=02r=a3a=23rd2Adr2=3×2π9+2π>0Therefore at a=23r we will have minimaSum of areas A is minimum when a=23r

​the sum of perimeters of an equilateral triangle and a circle is 'k'(constant). Prove that their combined area is minimum when the side of the triangle is 2root3 times the radius of the circle.

the sum of perimeters of an equilateral triangle and a circle is 'k'(constant). Prove that their combined area is minimum when the side of the triangle is 2root3 times the radius of the circle.