Question

Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3 cm and 3.0005 cm, respectively.

Answer 1

Let internal radius = r and external radius = R
$\therefore V=\frac{4}{3}\pi \left[\right(3.0005{)}^3-\left(3{)}^3\right]$
Now we shall use differentiation to get approximate value of $(3.0005{)}^3.$
Let $(3.0005{)}^3=y+△y$
and x = 3, $△x=0.0005$
Also, let $y=x^3$
On differentiating both sides w.r.t, x , we get
$\frac{dy}{dx}=3x^2\therefore △y=\frac{dy}{dx}\times △x=3x^2\times 0.0005=3\times 3^2\times 0.0005=27\times 0.0005=0.0135$
Also, $(3.0005{)}^3=y+△y=3^3+0.0135=27.0135\therefore V=\frac{4}{3}\pi [27.0135-27.000]=\frac{4}{3}\pi [0.0135]=4\pi \times (0.0045)=0.0180\pi c{m}^3$