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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2021 · Mathematics · Application of Derivatives

A 24 cm long wire is bent to form a triangle with one of the angles...

A 24 cm long wire is bent to form a triangle with one of the angles as 60∘60^{{}^{\circ}}. What is the altitude of the triangle having the greatest possible area?

Options

  1. A43cm4 \sqrt{3} \mathrm{ cm}
  2. B23cm2 \sqrt{3} \mathrm{ cm}
  3. C6cm6 cm
  4. D3cm3 cm

Correct answer

A. 43cm4 \sqrt{3} \mathrm{ cm}

Step-by-step solution

Among all triangles that have same perimeter, equilateral triangles are one that enclose maximum area.
243=8cm\frac{24}{3}=8 \mathrm{ cm} Here, AB=BC=CA=243=8cmA B=B C=C A=\frac{24}{3}=8 \mathrm{ cm} Area of △ABC=34(BC)2=34(8)2=163cm2AD\triangle A B C=\frac{\sqrt{3}}{4}(B C)^2=\frac{\sqrt{3}}{4}(8)^2=16 \sqrt{3} \mathrm{ cm}^2A D is the altitude of the △ABC\triangle A B C. Then, Area of △ABC=12×BC×AD⇒163=12×8×AD∴AD=43cm\triangle A B C=\frac{1}{2} \times B C \times A D\Rightarrow 16 \sqrt{3}=\frac{1}{2} \times 8 \times A D\therefore A D=4 \sqrt{3} \mathrm{ cm}
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