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

Internal radius r = 3 cm

And external radius R = r + Δr = 3.0005 cm

∴ Δr = 3.0005 – 3 = 0.0005 cm

Let y = r³

⇒ y + Δy = (r + Δr)³

= R³

= (3.0005)³   ......(i)

Differentiating both sides w.r.t., r, we get

`"dy"/"dr"` = 3r²

∴ Δy = `"dy"/"dr" xx Δ"r"` = 3r² × 0.0005

= 3 × (3)² × 0.0005

= 27 × 0.0005

= 0.0135

∴ (3.0005)³  = y + Δy   .....[From equation (i)]

= (3)³ + 0.0135

= 27 + 0.0135

= 27.0135

Volume of the shell = `4/3 pi ["r"^3 - "r"^3]`

= `4/3 pi [27.0135 - 27]`

= `4/3 pi xx 0.0135`

= 4π × 0.005

= 4 × 3.14 × 0.0045

= 0.018π cm³

Hence, the approximate volume of the metal in the shell is 0.018π cm³.