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