Name: Second Derivative Test
Uploaded: 2019-07-19
Duration: 12 min 48 s
Description: The maxima and minima of a function are the largest and smallest value of the function. Watch this amazing video to find the maxima and minima of a function.

Question

A chord of length

R

divides a circular area of radius

R

into two regions. Find the sides of the rectangle with the largest area that can be inscribed in the smaller region with one side along the given chord.

Accepted Answer

Let $U V W X$ be inscribed rectangle,

$E$ be midpoint of $W X$ and

$F$ be midpoint of $U V$

Given, chord $P Q$ of length $R$

In $∆ F O Q, O Q = R$ and $F Q = \frac{R}{2}$

So, $O F = \sqrt{R^{2} - {(\frac{R}{2})}^{2}} = \frac{\sqrt{3}}{2} R$ (using Pythagoras Theorem)

Again, let $∠ E O W = θ$

$\Rightarrow O E = R \cos θ$ and $E W = R \sin θ$

And, $E F = O E - O F$

$\Rightarrow E F = R (\cos θ - \frac{\sqrt{3}}{2})$

Let area of rectangle $U V W X = A (θ)$

$\Rightarrow A (θ) = F E . W X$

$\Rightarrow A (θ) = 2 R^{2} \sin θ (\cos θ - \frac{\sqrt{3}}{2})$

$\Rightarrow A^{'} (θ) = 2 R^{2} [\cos^{2} θ - \cos θ \frac{\sqrt{3}}{2} + \sin θ (- \sin θ)]$

For extrema, $A^{'} (θ) = 0$

$\Rightarrow \cos^{2} θ - \frac{\sqrt{3}}{2} \cos θ - \sin^{2} θ = 0$

$\Rightarrow 4 \cos^{2} θ - \sqrt{3} \cos θ - 2 = 0$

$\Rightarrow \cos θ = \frac{\sqrt{3} \pm \sqrt{35}}{8}$

Since $θ$ is acute $\Rightarrow \cos θ > 0$

$\Rightarrow \cos θ = \frac{\sqrt{3} + \sqrt{35}}{8}$

Also, $f^{''} (θ) < 0$ , for $\cos θ = \frac{\sqrt{3} + \sqrt{35}}{8}$

So, $A (θ)$ will be maximum.

Sides of rectangle are $U V,$

$U V = 2 R \sin θ$

$\Rightarrow U V = 2 R \sqrt{1 - {(\frac{\sqrt{3} + \sqrt{35}}{8})}^{2}}$

$U X = R \cos θ - R \frac{\sqrt{3}}{2}$

$\Rightarrow U X = R [\frac{\sqrt{3} + \sqrt{35}}{8} - \frac{\sqrt{3}}{2}]$

$\Rightarrow U X = R [\frac{\sqrt{35} - 3 \sqrt{3}}{8}]$

A chord of length $R$ divides a circular area of radius $R$ into two regions. Find the sides of the rectangle with the largest area that can be inscribed in the smaller region with one side along the given chord.

Important Questions on Monotonicity, Maxima and Minima

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