Question

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

• A. $0.180\, \pi cm ^3$
• B. $0.0180\, \pi cm ^3$
• C. $1.080 \,\pi cm ^3$
• D. $1.0180 \,\pi cm ^3$

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 $
$\Rightarrow \frac{d y}{d x}=3 x^2$
$ \therefore \Delta y=\frac{d y}{d x} \times \Delta x=3 x^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
$=0.0180 \pi cm ^3$