If a rectangle is inscribed in an equilateral triangle of side length 2 2 as shown in the figure, then the square of the largest area of such a rectangle is _____.

In Δ D B F , tan 60 ° = 2 b 2 2 - ℓ ⇒ b = 3 2 2 - ℓ 2 A = Area of rectangle = ℓ × b ⇒ A = ℓ × 3 2 2 2 - ℓ On differentiating, we get d A d ℓ = 3 2 2 2 - ℓ - ℓ · 3 2 = 0 ⇒ ℓ = 2 Now, A = ℓ × b = 2 × 3 2 2 = 3 ⇒ A 2 = 3 .

Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMaxima-MinimaMedium2 minPYQ_2021

MathematicsMediumnumerical

If a rectangle is inscribed in an equilateral triangle of side length $2 \sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is _____.

Answer:

3.00

Solution:

In $Δ D B F,$

$\tan 60 ° = \frac{2 b}{2 \sqrt{2} - ℓ}$

$\Rightarrow b = \frac{\sqrt{3} (2 \sqrt{2} - ℓ)}{2}$

$A =$ Area of rectangle

$= ℓ \times b$

$\Rightarrow A = ℓ \times \frac{\sqrt{3}}{2} (2 \sqrt{2} - ℓ)$

On differentiating, we get

$\frac{d A}{d ℓ} = \frac{\sqrt{3}}{2} (2 \sqrt{2} - ℓ) - \frac{ℓ \cdot \sqrt{3}}{2} = 0$

$\Rightarrow ℓ = \sqrt{2}$

Now, $A = ℓ \times b$

$= \sqrt{2} \times \frac{\sqrt{3}}{2} (\sqrt{2}) = \sqrt{3}$

$\Rightarrow A^{2} = 3 .$

Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Maxima-Minima

⏱ 2mℹ️ Source: PYQ_2021

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