Question

Q3......i am posting induvidual Questions that i cannot solve

Answer 1

$Given:\frac{dS}{S}=\alpha \\We.know.that\\V=\frac{4\pi r^3}{3}\\so,\frac{dV}{dr} = 4\pi r^2\\\frac{dV}{dr*V}= \frac{4\pi r^2}{\frac{4\pi r^3}{3}}\\Simplify,\\\frac{dv}{vdr}=\frac{3}{r}\\\frac{dv}{v}=\frac{3}{r}dr ...... (1st)\\We.know.that\\S=4\pi r^2\\\frac{dS}{dr}=8\pi r\\\frac{dS}{dr*S}=\frac{8\pi r}{4\pi r^2}\\\frac{dS}{dr*S}=\frac{2}{r}\\\frac{dS}{S}=\frac{2}{r}dr\\Rearrange .above .to .get:\\\frac{dr}{r}=\frac{dS}{S*2}=\frac{\alpha }{2}$

$Putting. this.relation.in.(1st):\\\\\frac{dv}{V}=\frac{3}{r}dr\\\\\frac{dv}{V}=\frac{3\alpha}{2}$

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