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

Explanation

Let internal radius = r and external radius = R
`therefore` Volume of hollow sphericla shell, `V=4/3pi[(3.0005)^(3) - (3)^(3)]`
Now, we shall use differentiation to get approximate value of `(3.0005)^(3)`.
Let `(3.0005)^(3)= y+Deltay`
and `x=3, Deltax=0.0005`
Also, let `y=x^(3)`
On differentiating both sides w.r.t. x, we get
`(dy)/(dx) = 3x^(2)`
`(Deltay) =(dy)/(dx) xx Deltax=3x^(2) xx 0.0005`
`=3 xx 3^(2) xx 0.0005`
`=27 xx 0.0005 = 0.00135`
Also,
`(3.0005)^(3) = y+ Deltay`
`=3^(3) + 0.0135 = 27.0135`
`therefore V=4/3pi[27.0135- 27.000]` [Using Eq. (i)]
`=4/3pi[0.0135]=4pi xx (0.0045)`
`=0.0180 pi cm^(3)`