The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.
Let r be the radius of the sphere, x be the side of the cube and S be the sum of the surface area of both. Then,
$S=4 \pi r^{2}+6 x^{2}$
$\Rightarrow x=\left(\frac{S-4 \pi r^{2}}{6}\right)^{\frac{1}{2}}$ .....(1)
Sum of volumes, $V=\frac{4}{3} \pi r^{3}+x^{3}$
$\Rightarrow V=\frac{4 \pi r^{3}}{3}+\left[\frac{\left(S-4 \pi r^{2}\right)}{6}\right]^{\frac{3}{2}}$ [From eq. (1)]
$\Rightarrow \frac{d V}{d r}=4 \pi r^{2}-2 \pi r\left[\frac{\left(S-4 \pi r^{2}\right)}{6}\right]^{\frac{1}{2}}$
For the minimum or maximum values of $V$, we must have
$\frac{d V}{d r}=0$ ....(2)
$\Rightarrow 4 \pi r^{2}-2 \pi r\left[\frac{\left(S-4 \pi r^{2}\right)}{6}\right]^{\frac{1}{2}}=0$ [From eq. (2)]
$\Rightarrow 4 \pi r^{2}=2 \pi r\left[\frac{\left(S-4 \pi r^{2}\right)}{6}\right]^{\frac{1}{2}}$
$\Rightarrow 4 \pi r^{2}=2 \pi r x$ [From eq. (1)]
$\Rightarrow x=2 r$
Now,
$\frac{d^{2} V}{d r^{2}}=8 \pi r-2 \pi\left[\frac{\left(S-4 \pi r^{2}\right)}{6}\right]^{\frac{1}{2}}-\frac{2 \pi r}{2}\left[\frac{\left(S-4 \pi r^{2}\right)}{6}\right]^{-\frac{1}{2}} \frac{(-8 \pi r)}{6}$
$\Rightarrow \frac{d^{2} V}{d r^{2}}=8 \pi r-2 \pi\left[\frac{\left(S-4 \pi r^{2}\right)}{6}\right]^{\frac{1}{2}}+\frac{4}{3} \pi^{2} r^{2}\left[\frac{6}{\left(S-4 \pi r^{2}\right)}\right]^{\frac{1}{2}}$
$\Rightarrow \frac{d^{2} V}{d r^{2}}=8 \pi r-2 \pi x+\frac{4}{3} \pi^{2} r^{2} \frac{1}{x}=8 \pi r-4 \pi r+\frac{2}{3} \pi^{2} r$
$\Rightarrow \frac{d^{2} V}{d r^{2}}=4 \pi r+\frac{2}{3} \pi^{2} r>0$
So, volume is minimum when x = 2r.
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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,