Question

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Answer 1

Question :

Find the value of 's' from following equation :

$\star\sf \ \ 75 = \dfrac{200}{200 + 20s} + \dfrac{1}{s}$

Solution :

$: \implies\sf 75 = \dfrac{200}{200 + 20s} + \dfrac{1}{s} \\\\ \\ : \implies\sf 75 = \dfrac{200s + 200 + 20s}{s(200 + 20s)} \\\\ \\ : \implies\sf 75 = \dfrac{220s + 200}{200s + 20s^2} \\\\ \\ : \implies\sf 75(200s + 20s^2) = 220s + 200 \\\\ \\ : \implies\sf 15000s + 1500s^2 = 220s + 200 \\\\ \\ : \implies\sf 1500s^2 + 15000s - 220s - 200 = 0 \\\\ \\ : \implies\sf 1500s^2 + 14780s - 200 = 0 \\\\ \\ : \implies\sf 4(375s^2 + 3695s - 50) = 0 \\\\ \\ : \implies\sf 375s^2 + 3695s - 50 = 0$

Now using quadratic formula :

$: \implies\sf s = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\\\ \\ : \implies\sf s = \dfrac{-3695 \pm \sqrt{(3695)^2 - 4 \times 375 \times (-50)}}{2 \times 375} \\\\ \\ : \implies\sf s = \dfrac{-3695 \pm \sqrt{13728025}}{750} \\\\ \\ : \implies\sf s = \dfrac{-3695 \pm 3705.13}{750} \\\\ \\ : \implies\sf s = \dfrac{-3695 + 3705.13}{750} \ \ or, \ \ s = \dfrac{-3695 - 3705.13}{750} \\\\ \\ : \implies\sf s = \dfrac{10.13}{750} \ \ or, \ \ s = \dfrac{-7400.13}{750}$

$: \implies\underline{\boxed{\sf{s = 0.014}}} \ \ \sf or, \ \ \underline{\boxed{\sf{s = -9.87}}} \qquad\quad [{\bold{Required \ answer}}]$

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

What is be the value of 's' in the following equation ?

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♣ ɢɪᴠᴇɴ ᴇQᴜᴀᴛɪᴏɴ :

$\sf{75=\dfrac{200}{200+20s}+\dfrac{1}{s}}$

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♣ ᴀɴꜱᴡᴇʀ :

$\boxed{\sf{s=\dfrac{-739+\sqrt{549121}}{150},\:s=\dfrac{-739-\sqrt{549121}}{150}}\left(\mathrm{Decimal}:\quad s=0.01351\dots ,\:s=-9.86684\dots \right)}$

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♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

$\boxed{\bf{75=\dfrac{200}{200+20s}+\dfrac{1}{s}}}$

Firstly Simplify  $\sf{\dfrac{200}{200+20s}}$  :

$\implies\sf{\dfrac{200}{200+20 s}= \quad \dfrac{10}{10+s}}$

$\sf{75=\dfrac{10}{10+s}+\dfrac{1}{s}}$

Find Least Common Multiplier of  10+s, s :

L.C.M Defination :

$\boxed{\star\:\:\mathrm{The\:LCM\:of\:}a,\:b\:\mathrm{is\:the\:smallest\:multiplier\:that\:is\:divisible\:by\:both\:}a\mathrm{\:and\:}b\:\:\star}$

$\text { Compute an expression comprised of factors that appear either in } 10+s \text { or } s$

$\sf{=s\left(s+10\right)}$

$\mathrm{Multiply\:by\:LCM=}s\left(s+10\right)$

$\sf{75s\left(s+10\right)=\dfrac{10}{10+s}s\left(s+10\right)+\dfrac{1}{s}s\left(s+10\right)}$

Simplify :

$\sf{75s\left(s+10\right)=11s+10}$

On solving $\sf{75s\left(s+10\right)=11s+10}$  we get $\sf{75s^2+739s-10=0}$

Solve with the quadratic formula :

Quadratic Equation Formula :

$\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}$

$\sf{x_{1,\:2}=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}}$

$\mathrm{For\:}\quad a=75,\:b=739,\:c=-10:\quad s_{1,\:2}=\dfrac{-739\pm \sqrt{739^2-4\cdot \:75\left(-10\right)}}{2\cdot \:75}$

$\sf{s=\dfrac{-739+\sqrt{739^{2}-4 \cdot 75(-10)}}{2 \cdot 75}= \dfrac{-739+\sqrt{549121}}{150}}$

$\sf{s=\dfrac{-739-\sqrt{739^{2}-4 \cdot 75(-10)}}{2 \cdot 75}= \dfrac{-739-\sqrt{549121}}{150}}$

$\mathrm{The\:solutions\:to\:the\:quadratic\:equation\:are:}$

$\boxed{\sf{s=\dfrac{-739+\sqrt{549121}}{150},\:s=\dfrac{-739-\sqrt{549121}}{150}}}$

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♣ ᴠᴇʀɪꜰʏ ꜱᴏʟᴜᴛɪᴏɴꜱ :

Find undefined (singularity) points:  s = -10, s = 0

Combine undefined points with solutions:

$\boxed{\sf{s=\dfrac{-739+\sqrt{549121}}{150},\:s=\dfrac{-739-\sqrt{549121}}{150}}}$

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