Calculus!!?

Question

The sum of the perimeters of an equilateral triangle and a square is 9. Find the dimensions of the triangle and the square that produce a minimum total area.

Answer 1

Side of equilateral triangle = x

The perimeters of an equilateral triangle = 3x

Side of a square = y

The perimeters of a square = 4y

3x + 4y = 9

4y = 9 - 3x

y = (9 - 3x)/4

Area of an equilateral triangle  = (√3/4)x²

Area of a square = y²

Total Area A = Area of an equilateral triangle + Area of a square

A = (√3/4)x² + y²

A = (√3/4)x² + (9 - 3x)²/16

To have minimum area , apply derivative both sides.

dA/dx = (d/dx) [ (√3/4)x² + (9 - 3x)²/16 ]

dA/dx = (√3/4)(d/dx)x² + (1/16)(d/dx)(9 - 3x)²

dA/dx = (√3/4)2x + (1/16)2(9 - 3x)(-3)

dA/dx = (√3/2)x - (3/8)(9 - 3x)

But area is constant : dA/dx = 0

(√3/2)x - (3/8)(9 - 3x) = 0

(4√3)x = (3)(9 - 3x)

(4√3)x = 27 - 9x

(4√3)x + 9x= 27

[(4√3)+9] x = 27

x = 27 / [(4√3)+9]

x = 1.695

y = (9 - 3x)/4 ⇒ y = (9 - 3*1.695)/4

y = 3.915/4

y = 0.97875

Side of equilateral triangle is 1.695 and Side of square is 0.97875

that produce a minimum total area.