Question

Based on the proportions given in the image, a rectangular gate with a triangular vent of equal sides needs to be constructed. If the area of the vent needs to be 4√3 m², what will be boundary length (in m) of the entire structure?​

Answer 1

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$\underline{\textsf{\textbf{\purple{Question\::}}}}$

Based on the proportions given in the image, a rectangular gate with a triangular vent of equal sides needs to be constructed. If the area of the vent needs to be 4√3 m² , what will be boundary length (in m) of the entire structure?

$\underline{\textsf{\textbf{ \purple{Given\::}}}}$

• Side of triangular vent = a m
• Area of triangular vent = 4√3 m²
• Length of rectangular gate = 7a m
• Breadth of gate = side of vent = a m

$\underline{\textsf{\textbf{ \purple{To\:find\::}}}}$

• Length of boundary (in m) of the entire structure?

$\underline{\textsf{\textbf{ \purple{Solution\::}}}}$

★ Using formula of area of equilateral ㅤtriangle :

$\qquad:\implies\:\bf{ Area_{(Equilateral\:triangle)}\:=\: \red{\dfrac{\sqrt{3}}{4}(Side)^2}}$

★ Values that we have :

• Area(Equilateral triangle) = 4√3 m²
• Side = a m

★ Putting all values in the formula :

$\qquad:\implies\:\sf 4\:\times\:\sqrt{3} = \dfrac{\sqrt{3}}{4}\:\times\:(a)^2$

$\qquad:\implies\:\sf 4\:\times\:\sqrt{3}\:\times\:\dfrac{4}{\sqrt{3}} = a^2$

$\qquad:\implies\:\sf 4\:\times\:\cancel{\sqrt{3}}\:\times\:\dfrac{4}{\cancel{\sqrt{3}}} = a^2$

$\qquad:\implies\:\sf 4 = a^2$

$\qquad:\implies\:\sf \sqrt{4} = a$

$\qquad:\implies\:\sf \sqrt{\underline{2\:\times\:2}} = a$

$\qquad:\implies\:\bf {a = \red{4\:m}}$

Therefore,

~Side(vent) = breadth(gate) = a m = 4 m

~Length(gate) = 7a m = (7 × 4) m = 28 m

★ Finding the length of boundary of ㅤㅤㅤentire structure :

➠ Length(boundary) = Perimeter(vent) + ㅤPerimeter(gate)

➠ Length(boundary) = (3×side) + [2(l+b)]

➠ Length(boundary) = (3×4) + [2(28+4)]

➠ Length(boundary) = 12 + 2(32)

➠ Length(boundary) = 12 + 64

➠ Length(boundary) = 76 m

This is the required answer.

$\small\underline{\boxed{\sf{Length\:of\:boundary\:of\:entire\:structure\:=\:\rm\purple{76\:m}}}}$

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