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QJEEM22Ma9c0f: JEE Main 2022 Mathematics - Class 12 JEEM22Ma9c0f Solution: A wire of length 22 m is to be cut into two pieces. One of the…

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is :

Multiple Choice Options

A 229+43
B 669+43
C 224+93
D 664+93

Step-by-Step Verified Solution

JEE Main 2022 (Online) 29th June Morning Shift Mathematics - Application of Derivatives Question 84 English Explanation

$4 a + 3 b = 22$

Total area $= A = a^{2} + \frac{\sqrt{3}}{4} b^{2}$

$A = {(\frac{22 - 3 b}{4})}^{2} + \frac{\sqrt{3}}{4} b^{2}$

$\frac{d A}{d B} = 2 (\frac{22 - 3 b}{4}) (\frac{- 3}{4}) + \frac{\sqrt{3}}{4} . 2 b = 0$

$\Rightarrow \frac{\sqrt{3 b}}{2} = \frac{3}{8} (22 - 3 b)$

$\Rightarrow 4 \sqrt{3} b = 66 - 9 b$

$∴ b = \frac{66}{9 + 4 \sqrt{3}}$

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