Question

Solve this:-

$( \frac{r(\pi + r)}{4})^{2} - \frac{\pi {r}^{2} }{2} = 4$
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Answer 1

[tex](\frac{r(\pi + r)}{4})^{2} - \frac{\pi {r}^{2} }{2} = 4 \\  \\ \implies \: ( \frac{r\pi  +  {r}^{2} }{4} )^{2}   - \frc{\pi {r}^{2} }{2} = 4 \\  \\ \implies \: \frac{({r\pi  +  {r}^{2} ) ^{2} }}{ {4}^{2} }  - \frac{\p {r}^{2} }{2} = 4  \\  \\  \sf\red {Using  \: the \:  Algeric \:  Identity  \: (a + b)² \:  =  \: a² + b² + 2ab}\\  \\   \implies \:  \frac{( {\pi{r})}^{2} \:  +  \: ( { {r}^{2}) }^{2}  \:  + (2 \times \pi{r} \:  \times  {r}^{2}) }{6}  - \frac{\pi {r}^{2} }{2} = 4   \\  \\  \implies \: \frac{  {\pi}^{2}  {r}^{2}  \:  \:  +  \:  {r}^{4}  \:  +  \: 2\pi {r}^{3} }{16}  \:  - \frac{\pi {r}^{2} }{2} = 4   \\  \\  \implies \: \frac{({\pi}^{2}  {r}^{2}  \:  \:  +  \:  {r}^{4}  \:  +  \: 2\pi {r}^{3} )\:  - 8\pi {r}^{2} }{16}  \:  =  \: 4 \\  \\  \imlies \:  \frac{ {r}^{4} \:  +{\pi}^{2}  {r}^{2} + 2\pi {r}^{3} \:  - 8\p {r}^{2}   }{16}  \:  =  \: 4 \\  \\  \implies \:  {r}^{4} \:  +{\pi}^{2}  {r}^{2} + 2\pi {r}^{3} \:  - 8\pi {r}^{2}    \:  = 4 \time 16 \\  \\  \implies \:  {r}^{4} \:  +{\pi}^{2}  {r}^{2} + 2\pi {r}^{3} \:  - 8\pi {r}^{2}    \:  = \: 64 \\  \\  \implies \:  {r}^{2} ( {r}^{2}  \:  +  \:  {\pi}^{2}  \:  + 2\pi{r} \:  - 8\pi)  \: =  \: 64 [tex]

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\sf{\left(\dfrac{r\left(\pi +r\right)}{4}\right)^2-\dfrac{\pi \:r^2}{2}=4}$

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♣ ᴀɴꜱᴡᴇʀ :

$\sf{\left(\dfrac{r\left(\pi +r\right)}{4}\right)^2-\dfrac{\pi r^2}{2}=4\quad :\quad r\approx \:-8.24781\dots ,\:r\approx \:2.68834\dots }$

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♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

$\left(\dfrac{r\left(\pi +r\right)}{4}\right)^2-\dfrac{\pi r^2}{2}=4$

$\implies\left(\dfrac{r\left(\pi +r\right)}{4}\right)^2\cdot \:2-\dfrac{\pi r^2}{2}\cdot \:2=4\cdot \:2$

$\implies\dfrac{r^2\left(r+\pi \right)^2}{8}-\pi r^2=8$

$\implies\dfrac{r^4}{8}+\dfrac{\pi r^3}{4}+\dfrac{\pi ^2r^2}{8}-\pi r^2=8$

$\implies\dfrac{r^4}{8}+\dfrac{\pi r^3}{4}+\dfrac{\pi ^2r^2}{8}-\pi r^2-8=8-8$

$\implies\dfrac{1\cdot \:r^4}{8}+\dfrac{\pi r^3}{4}+\left(\dfrac{\pi ^2}{8}-\pi \right)r^2-8=0$

$\text{Find one solution for }0.125 r^{4}+0.78539 \ldots r^{3}-1.90789 \ldots r^{2}-8=0 using Newton - Raphson r \approx-8.24781$

$\begin{aligned}&\text { Find one solution for } 0.125 r^{3}-0.24557 \ldots r^{2}+0.11760 \ldots r-0.96995 \ldots=0\\&\text { using Newton - Raphson: } r \approx 2.68834 \ldots\end{aligned}$

$\mathrm{The\:solutions\:are}$

$r\approx \:-8.24781\dots ,\:r\approx \:2.68834\dots$