Question

The relative error in the determination of the surface area of a sphere is $\alpha$. Then the relative error in the determination of its volume is :

• A. $\frac{2}{3}\alpha$
• B. $\frac{5}{2}\alpha$
• C. $\frac{3}{2}\alpha$
• D. $\alpha$

Answer 1

The correct option is C $\frac{3}{2}\alpha$

S=$4\pi R^2$

$\ln S$= $\ln \left(4\pi \right)+\ln \left(R^2\right)$

$\ln S=2\ln R$

$\frac{\Delta S}{S}=2\frac{\Delta R}{R}=\alpha$

$\frac{\Delta R}{R}=\frac{\alpha }{2}$ ------------(1)

V= $\frac{4}{3}\pi R^3$

$\ln V$ = $\ln \left(\frac{4}{3}\pi \right)+\ln R^3$

$\ln V=3\ln R$

$\frac{\Delta V}{V}=3\frac{\Delta R}{R}$

$\frac{\Delta V}{V}=3\left(\frac{\alpha }{2}\right)$