How do you find the dimensions of the rectangle with largest area that can be inscribed in a semicircle of radius $r$ $r$ ?

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How do you find the dimensions of the rectangle with largest area that can be inscribed in a semicircle of radius $r$ $r$ ?

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Answer 1 · 2018-08-12T06:05:09

The dimensions of the rectangle is $\sqrt{2} r$ $sqrt2r$ and $\frac{r}{\sqrt{2}}$ $r/sqrt2$

Explanation:

The equation of the semicircle is

$x^{2} + y^{2} = r^{2}$ $x^2+y^2=r^2$ ....................... $(1)$ $(1)$

The area of the rectangle is

$A = 2 x y$ $A=2xy$ .................... $(2)$ $(2)$

From equation $(1)$ $(1)$ , we get

$y^{2} = r^{2} - x^{2}$ $y^2=r^2-x^2$

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

Plugging this value in equation $(2)$ $(2)$

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

Differentiating wrt $x$ $x$ using the product rule

$\frac{d A}{d x} = 2 \sqrt{r^{2} - x^{2}} - 2 \frac{x^{2}}{\sqrt{r^{2} - x^{2}}}$ $(dA)/dx=2sqrt(r^2-x^2)-2x^2/sqrt(r^2-x^2)$

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

$= \frac{2 r^{2} - 4 x^{2}}{\sqrt{r^{2} - x^{2}}}$ $=(2r^2-4x^2)/(sqrt(r^2-x^2))$

The critical points are when

$\frac{d A}{d x} = 0$ $(dA)/dx=0$

That is

$\frac{2 r^{2} - 4 x^{2}}{\sqrt{r^{2} - x^{2}}} = 0$ $(2r^2-4x^2)/(sqrt(r^2-x^2))=0$

$r^{2} = 2 x^{2}$ $r^2=2x^2$

$x = \frac{r}{\sqrt{2}}$ $x=r/sqrt2$

Then,

$y = \sqrt{r^{2} - x^{2}} = \sqrt{r^{2} - \frac{r^{2}}{2}} = \frac{r}{\sqrt{2}}$ $y=sqrt(r^2-x^2)=sqrt(r^2-r^2/2)=r/sqrt2$

The maximum area is

$A = 2 \cdot \frac{r}{\sqrt{2}} \cdot \frac{r}{\sqrt{2}} = r^{2}$ $A=2*r/sqrt2*r/sqrt2=r^2$

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How do you find the dimensions of the rectangle with largest area that can be inscribed in a semicircle of radius $r$ $r$ ?

1 Answer

Explanation:

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Impact of this question

Science

Math

Humanities

... and beyond

How do you find the dimensions of the rectangle with largest area that can be inscribed in a semicircle of radius rr ?

1 Answer

Explanation:

Related questions

Impact of this question

How do you find the dimensions of the rectangle with largest area that can be inscribed in a semicircle of radius $r$ $r$ ?