The sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area.

Solution:

Consider an equilateral triangle of side x and a square with side y

The perimeter of the triangle will therefore = 3x-------->(1)

The perimeter of the square = 4y--------> (2)

The sum of the perimeter of the equilateral triangle and the square is 10. So we can write:

3x + 4y = 10-------->(3)

The area of the equilateral  triangle with side ‘x’ will be given by the following expression:

\(\sqrt{\frac{3x}{2}(\frac{3x}{2}-\frac{x}{2})(\frac{3x}{2}-\frac{x}{2})(\frac{3x}{2}-\frac{x}{2}))}\)

Where 3x/2 is the semi perimeter which is expressed as the sum of three sides divided by 2

Semi-perimeter = (x + x + x)/2 = 3x/2

Area of equilateral triangle = \(\sqrt{\frac{3x}{2}(\frac{x}{2})(\frac{x}{2})(\frac{x}{2})}\)

Area of the equilateral Triangle = \(\sqrt{\frac{3x^{4}}{16}}\)

Area of the equilateral Triangle = (√3/4)x²-------->(4)

The area of the square will be given as:

Area of the square = y × y = y²-------->(5)

Let A = sum of the area of the equilateral triangle and the area of the square

A =  (√3/4)x² +  y²-------->(6)

From equation (3) we have:

y = (10 - 3x) / 4-------->(7)

Substituting (7) in (6)

A = (√3/4)x²  + ((10 - 3x)/4)²

   =   (√3/4)x²  + (100 + 9x² - 60x) / 16

   =    (√3/4)x² + 25/4 + (9/16)x²  - (15/4)x

   =   (  (√3/4) + (9/16))x² - (15/4)x + (25/4)  -------->(8)

Differentiating (8) w.r.t. X we get

dA/dx = 2((√3/4) + (9/16))x -15/4  -------->(9)

Equating (9) to zero we have 

 2((√3/4) + (9/16))x -15/4 = 0

2((√3/4) + (9/16))x = 15/4

((√3/4) + (9/16))x  = 15/8

x =  15/(8((√3/4) + (9/16)) = 15/(2√3+ 9/2)

x = 30/(4√3 + 9) -------->(10) 

From (8) we have 

y = 10/4 - 3x/4 = 5/2 - (3x/4)

Substituting the value of x from (10) in equation above we have

y =  5/2  - (3/4)(30/(4√3 + 9))

   =  (10((4√3 + 9)) - 90) / 4(4√3 + 9) 

   =        (40√3 + 90 - 90 ) / 4(4√3 + 9) 

  =      (40√3) / 4(4√3 + 9)

  =   (10√3 )/ (4√3 + 9) -------->(11)

To confirm that the dimensions of the side of the triangle and that of the square produces the minimum area equation (9) is differentiated again:

d²A/dx² =  2((√3/4) + (9/16)) = √3/2 + 9/8

Since d²A/dx² is +ve the dimensions of x and y will result in minimum area of both the triangle and square.

---

The sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area.

Summary:

The dimensions of the triangle and the square that produce a minimum total area are given as  x = 30  / (4√3 + 9)    and y =  10√3 / (4√3 + 9)  respectively.