Question

The approximate volume of metal in a hollow spherical shell whose internal and external radii are $3cm$ and $3.0005cm$, respectively is

• A. $0.180\pi c{m}^3$
• B. $0.0180\pi c{m}^3$
• C. $1.080\pi c{m}^3$
• D. $1.0180\pi c{m}^3$

Answer 1

Answer: (B)

Let $r$ be the internal radius and $R$ be the external radius.
$\therefore$ Volume of the hollow spherical shell
$V=\frac{4}{3}\times \pi \left(R^3-r^3\right)=\frac{4}{3}\pi \left[(3.0005{)}^3-3^3\right]$
Let $(3.0005{)}^3=y+\Delta y$
Let $x=3\Delta x=0.0005$
Let $y=x^3$
$\Rightarrow \frac{dy}{dx}=3x^2$
$\therefore \Delta y=\frac{dy}{dx}\times \Delta x=3x^2\times 0.0005$
$=3\times 3^2\times 0.0005=0.0135$
$\because (3.0005{)}^3=y+\Delta y$
$=3^3+0.0135=27.0135$
$\therefore V=\frac{4}{3}\pi (27.0135-27.000)$
$=\frac{4}{3}\pi \times 0.0135$
$=4\pi \times 0.0045$
$=0.0180\pi$
$\therefore$ Approximate volume of the spherical shell
$=0.0180\pi c{m}^3$