A rectangle ABCD is inscribed in the region b | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The rectangle is symmetric about the line $x = \frac{\pi}{2}$.Let the distance from the center $x = \frac{\pi}{2}$ to the sides be $\alpha$.The $x

Vedclass · Accepted Answer

$\cot \alpha = \alpha$

Vedclass · Answer

(B) The rectangle is symmetric about the line $x = \frac{\pi}{2}$.Let the distance from the center $x = \frac{\pi}{2}$ to the sides be $\alpha$.The $x-$coordinates of the vertices $B$ and $C$ are $\frac{\pi}{2} - \alpha$ and $\frac{\pi}{2} + \alpha$ respectively.The height of the rectangle is $y = \sin\left(\frac{\pi}{2} + \alphaight) = \cos \alpha$.The width of the rectangle is $(\frac{\pi}{2} + \alpha) - (\frac{\pi}{2} - \alpha) = 2\alpha$.Thus,the area $A$ of the rectangle is $A = (2\alpha)(\cos \alpha) = 2\alpha \cos \alpha$.To find the maximum area,we differentiate $A$ with respect to $\alpha$:$\frac{dA}{d\alpha} = 2(\cos \alpha - \alpha \sin \alpha)$.Setting $\frac{dA}{d\alpha} = 0$ gives $\cos \alpha = \alpha \sin \alpha$,which simplifies to $\cot \alpha = \alpha$ (since $\alpha 
eq 0$ and $\sin \alpha 
eq 0$ in the interval $(0, \frac{\pi}{2})$).

$A$ rectangle $ABCD$ is inscribed in the region bounded by $y = \sin x$ and the $x-$axis for $x \in [0, \pi]$ (as shown in the figure). The area of the rectangle is maximum when $'\alpha'$ satisfies:

Similar Questions

If $x$ is real and $\alpha, \beta$ are maximum and minimum values of $\frac{x^2-x+1}{x^2+x+1}$ respectively,then $\alpha+\beta=$

$A$ YouTube short video is going viral according to the function $f(t) = -2t^3 + 3t^2 + 5$. At what time $t$ (in hours) does the video get the maximum number of shares?

The function $f$ is defined by $f(x) = x^p (1 - x)^q$ for all $x \in R$,where $p, q$ are positive integers. The function has a maximum value for $x$ equal to:

$A$ stone moving vertically upwards has its equation of motion $s = 490t - 4.9t^2$. The maximum height reached by the stone is

Vedclass Test Series

Exam Paper Generator

Online Exam Module