Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3 cm and 3.0005 cm, respectively.
Given,
The internal radius r = 3 cm
And, external radius R = r + ∆r =3.0005 cm
∆r = 3.0005 – 3 = 0.0005 cm
Let y = r3 ⇒ y + ∆y = (r + ∆r)3 = R3 = (3.0005)3
Differentiating both sides w.r.t., r, we get
$\frac{d y}{d r}=3 r^{2}$
So, $\Delta y=\frac{d y}{d r} \times \Delta r=3 r^{2} \times 0.0005$
$=3 \times(3)^{2} \times 0.0005=27 \times 0.0005=0.0135$
$\therefore(3.0005)^{3}=y+\Delta y$
Volume of the shell $=\frac{4}{3} \pi\left[R^{3}-r^{3}\right]$
$=\frac{4}{3} \pi[27.0135-27]=\frac{4}{3} \pi \times 0.0135$
$=4 \pi \times 0.005=4 \times 3.14 \times 0.0045=0.018 \pi \mathrm{cm}^{3}$
Therefore, the approximate volume of the metal in the shell is 0.018π cm3
$1 \frac{2}{3}$
$5 \frac{1}{3}$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,