JEE Main 2022 · 30 Jul · Shift 2 · Mathematics Q84

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MathematicsInteger+4 / -1Question 84 of 90

An air duct is made of a thick metallic sheet. The duct is open from front and back. Its cross-section is a trapezium of base angles 60° and area 1.5 m $^2$ . Its length is 1 m. (see the figure). If its height $x=x_0$ (in metres) so that the area of the metallic sheet used for its construction is minimum, then $16(x_0)^4$ is equal to _______.

A duct 1 m long whose cross-section is an isosceles trapezium with base angles 60° and height x, open at the front and back.

Your answer

Answer and worked solution

Official answer: 27

Perimeter. With height x and top width b, the bottom is $b+\frac{2x}{\sqrt3}$ and each slant side $\frac{2x}{\sqrt3}$ . Area $\left(b+\frac x{\sqrt3}\right)x=1.5$ gives $b=\frac{1.5}x-\frac x{\sqrt3}$ , and the sheet area (perimeter × 1 m) is $\frac3x+\frac{4x}{\sqrt3}$ .

Minimum at $x^2=\frac{3\sqrt3}4$ , so $16x_0^4=27$ .

Answer: 27

Topics:Maxima and minima

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JEE Main 2022 · 30 Jul · Shift 2, Mathematics question 84