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A rectangle is inscribed in a semi circle with radius $r$ with one of its sides at the diameter of the semi circle. Find the dimensions of the rectangle so that its area is a maximum.

My Try:

Let length of the side be $x$, Then the length of the other side is $2\sqrt{r^2 -x^2}$, as shown in the image.

Rectangle inscribed in a semi circle

Then the area function is

$$A(x) = 2x\sqrt{r^2-x^2}$$

$$\begin{align}A'(x) &= 2\sqrt{r^2-x^2}-\frac{4x}{\sqrt{r^2-x^2}}\\ &=\frac{2}{\sqrt{r^2-x^2}} (r^2 - 2x -x^2)\end{align}$$

setting $A'(x) = 0$,

$$\implies x^2 +2x -r^2 = 0$$

Solving, I obtained:

$$x = -1 \pm \sqrt{1+r^2}$$

That however is not the correct answer, I cannot see where I've gone wrong? Can someone point out any errors and guide me the correct direction. I have a feeling that I have erred in the differentiation.

Also how do I show that area obtained is a maximum, because the double derivative test here is long and tedious.

Thanks!

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You have dropped an $x$ in calculating your derivative. By applying the product rule: $$\begin{align}A'(x) &= 2x\left(\frac{1}{2}(r^2-x^2)^{-1/2}(-2\color{red}{x})\right) + 2\sqrt{r^2-x^2}\\ &= \frac{-2x^{\color{red}{2}}}{\sqrt{r^2-x^2}} + 2\sqrt{r^2-x^2}\end{align}$$

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  • $\begingroup$ Ah... I see, that was silly, thanks $\endgroup$ – GenAsis Jul 20 '14 at 3:06
  • $\begingroup$ @GeniusAsis No problem. :) $\endgroup$ – apnorton Jul 20 '14 at 3:06
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Let $\theta$ be the angle that the slanted red (?) line on the right makes with the horizontal.

Then the height of the rectangle is $r\sin\theta$ and the base is $2r\cos\theta$, for an area of $r^2\sin\theta\cos\theta$.

This is $\frac{r^2}{2}\sin 2\theta$. But $\sin 2\theta$ has a maximum value of $1$, at $\theta=\frac{\pi}{4}$.

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hint :$x\sqrt{r^2-x^2}=\sqrt{x^2(r^2-x^2)}\le \dfrac{x^2+(r^2-x^2)}{2}=\dfrac{r^2}{2}$

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enter image description here

Eqn of circle: x^2 + y^2 = r^2

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area of rectangle = a = 2xy

==> a^2 = 4 x^2 y^2

==> a^2 = 4 x^2 (r^2 - x^2)

==> a^2 = 4 (x^2 r^2 - x^4)

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differentiating:

==> 2a da/dt = 4 (2x r^2 - 4x^3)

since, da/dt=0

==> 2 x r^2 = 4 x^3

==> x = r/sqrt(2)

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second derivative test:

==>4 (2 r^2 - 12 x^2)

==>8 r^2 - 24 r^2

==>(-16) r^2

==> -ve

therefore, maximum area for x= r/sqrt(2)

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finding y:

==>y^2 = r^2 - x^2

==>y^2 = r^2 - (r^2)/2

==>y=r/sqrt(2)

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For convenience, think that the circle has its center at (0,0). We then consider the upper semicircle of x^2+y^2=a^2. (1)

The area of the inscribed rectangle would be A=2xy dA/dx=(2x)'y+2x(dy/dx) =2y+2x(dy/dx)

diff (1) (d/dx)(x^2+y^2)=(d/dx)(a^2) <=> 2x+2y(dy/dx)=0 <=> dy/dx = -x/y

Now, dA/dx = 2y+2x(-x/y) <=> dA/dx = 2(y^2-x^2)/y <=>(1) dA/dx = 2(a^2-2x^2)/y

dA/dx = 0 <=> a^2-2x^2 = 0 <=> x = a/sqrt(2)

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By elementary means:

The maximum of $2x\sqrt{r^2-x^2}$ is also that of $x^2\sqrt{r^2-x^2}$, by squaring, or that of $t(1-t)$, by setting $r^2t=x^2$.

The expression $t(1-t)$ describes a parabola, with roots at $t=0$ and $t=1$, and by symmetry its vertex must be at $t=\dfrac12$.

Hence the requested area,

$$2\frac r{\sqrt 2}\sqrt{r^2-\frac{r^2}2}=r^2.$$

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Doubling the semi-circle to obtain a full circle, we now have a rectangle inscribed in a circle, with area equal to twice the area of your starting rectangle. Letting $x$ be either angle formed by its diagonals, we need to maximize $$ A(x)=2r \sin x\,. $$ This has a maximum when the diagonals are orthogonal, i.e. when the rectangle is a square, with side $\sqrt 2\, r$. Hence, the dimensions of the original rectangle are $\sqrt2\,r$ and $r/\sqrt2\,$.

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