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.

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 )

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calculista calculista · Answer 2 · 2018-08-02T20:24:56+02:00

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 [tex] 10 [/tex]

Perimeter of triangle is equal to [tex] 3x [/tex]

Perimeter of the square is equal to [tex] 4y [/tex]

so

[tex] 3x+4y=10\\ 4y=10-3x [/tex]

[tex] y=2.5-0.75x [/tex] ------> equation [tex] 1 [/tex]

Find the area of equilateral triangle

Applying the law of sines

[tex] A1=\frac{1}{2} *x^{2} *sin 60\\ A1=\frac{\sqrt{3}}{4} *x^{2} [/tex]

Find the area of the square

[tex] A2=y^{2} [/tex]

Fin the total area

[tex] At=A1+A2 [/tex]

[tex] At=\frac{\sqrt{3}}{4} x^{2} +y^{2} [/tex] ----> equation [tex] 2 [/tex]

Substitute equation [tex] 1 [/tex] in equation [tex] 2 [/tex]

[tex] At=\frac{\sqrt{3}}{4} x^{2} +(2.5-0.75x)^{2} [/tex]

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 [tex] (1.88,2.72) [/tex]

that means that

for [tex] x=1.88 units [/tex] the total area is equal to[tex] 2.72 units^{2} [/tex] (is the minimum total area)

find the value of y

[tex] y=2.5-0.75*1.88 [/tex]

[tex] y=1.09 units [/tex]

therefore

the answer is

the length side of the equilateral triangle is equal to [tex] 1.88 units[/tex]

the length side of the square is equal to [tex] 1.09 units[/tex]