Question

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.

Answer 1

Let

x-------> the length side of the equilateral triangle

y-------> the length side of the square

we know that

The sum of the perimeters of an equilateral triangle and a square is
`10`

Perimeter of triangle is equal to
`3x`

Perimeter of the square is equal to
`4y`

so

`3x+4y=10\\ 4y=10-3x`

`y=2.5-0.75x` ------> equation
`1`

Find the area of equilateral triangle

Applying the law of sines

`A1=(1)/(2) *x^(2) *sin 60\\ A1=(√(3))/(4) *x^(2)`

Find the area of the square

`A2=y^(2)`

Fin the total area

`At=A1+A2`

`At=(√(3))/(4) x^(2) +y^(2)` ----> equation
`2`

Substitute equation
`1` in equation
`2`

`At=(√(3))/(4) x^(2) +(2.5-0.75x)^(2)`

Using a graph tool

see the attached figure

we know that

the vertex of the graph is the point with the minimum total area

the vertex of the graph is the point
`(1.88,2.72)`

that means that

for
`x=1.88 units` the total area is equal to
`2.72 units^(2)` (is the minimum total area)

find the value of y

`y=2.5-0.75*1.88`

`y=1.09 units`

therefore

the answer is

the length side of the equilateral triangle is equal to
`1.88 units`

the length side of the square is equal to
`1.09 units`

Answer 2

x - side of an equilateral triangle, y - side of a square;
3 x + 4 y = 10
4 y = 10 - 3 x
b = ( 10 - 3 x ) /4
Total area:
A = x ²√3/4 +y² = x²√3/4 + ( 10- 3 x )²/16 = x²√3/4 + 9 x²/16 - 15 x/4 + 25/4
A` = x√3/2 + 9 x/8 - 15/4
A` = 0
4 x√3 + 9 x = 30
15.92 x = 30
x = 30 : 14.92 ≈ 1.88
y = (10 - 1.88 ) : 4 ≈ 2.03
Answer: dimensions are x = 1.88 ( triangle ) and y = 2.03 ( square )