Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius $12 \mathrm{~cm}$ is $16 \mathrm{~cm}$.
Let the height, radius of base and volume of the cone be $h, r$ and $V$, respectively. Then,
$h=R+\sqrt{R^{2}-r^{2}}$
$\Rightarrow h-R=\sqrt{R^{2}-r^{2}}$
Squaring both the sides, we get
$h^{2}+R^{2}-2 h R=R^{2}-r^{2}$
$\Rightarrow r^{2}=2 h R-h^{2}$ ....(1)
Now,
$V=\frac{1}{3} \pi r^{2} h$
$\Rightarrow V=\frac{\pi}{3}\left(2 h^{2} R-h^{3}\right)$ $[$ From eq. (1) $]$
$\Rightarrow \frac{d V}{d h}=\frac{\pi}{3}\left(4 h R-3 h^{2}\right)$
For maximum or minimum values of $V$, we must have
$\frac{d V}{d h}=0$
$\Rightarrow \frac{\pi}{3}\left(4 h R-3 h^{2}\right)=0$
$\Rightarrow 4 h R=3 h^{2}$
$\Rightarrow h=\frac{4 R}{3}$
Now,
$\frac{d^{2} V}{d h^{2}}=\frac{\pi}{3}(4 R-6 h)$
$\Rightarrow \frac{\pi}{3}(4 R-8 R)=0$
$\Rightarrow \frac{-4 \pi R}{3}<0$
So, the volume is maximum when $h=\frac{4 R}{3}$.
$\Rightarrow h=\frac{4 \times 12}{3}=16 \mathrm{~cm}$
- 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,
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,