Prove that a conical tent of given capacity will require the least amount of canavas when the height is $\sqrt{2}$ times the radius of the base.
Let the surface area of conical tent be $S=\pi \mathrm{r} \sqrt{\mathrm{r}^{2}+\mathrm{h}^{2}}$
Let the volume of the conical tent $\mathrm{V}=\frac{1}{3} \pi \mathrm{r}^{2} \mathrm{~h}$
$\Rightarrow \mathrm{h}=\frac{3 \mathrm{~V}}{\pi \mathrm{r}^{2}}$
$\therefore S=\pi r \sqrt{r^{2}+\left(\frac{3 V}{\pi r^{2}}\right)^{2}}$
$\Rightarrow \mathrm{S}=\frac{1}{\mathrm{r}} \sqrt{\pi^{2} \mathrm{r}^{6}+9 \mathrm{~V}^{6}}$
Now differentiating with respect to $r$ we get,
$\frac{d S}{d r}=\frac{d}{d r}\left[\frac{1}{r} \sqrt{\pi^{2} r^{6}+9 V^{6}}\right]$
$=\frac{1}{r} \frac{6 \pi^{2} r^{5}}{2\left(\sqrt{\pi^{2} r^{6}+9 V^{6}}\right)}-\frac{\sqrt{\pi^{2} r^{6}+9 V^{6}}}{r^{2}}$
For minima putting $\frac{d S}{d r}=0$ we get,
$\frac{3 \pi^{2} r^{4}}{\sqrt{\pi^{2} r^{6}+g V^{6}}}=\frac{\sqrt{\pi^{2} r^{6}+g V^{6}}}{r^{2}}$
$\Rightarrow 3 \pi^{2} r^{6}=\pi^{2} r^{6}+9 V^{6}$
$\Rightarrow 2 \pi^{2} r^{6}=9 V^{6}$
Substitutting the value of $V$ we get,
$2 \pi^{2} r^{6}=9\left(\frac{1}{3} \pi r^{2} h\right)^{2}$
$\Rightarrow 2 \pi^{2} r^{6}=\pi^{2} r^{4} h^{2}$
$\Rightarrow 2 r^{2}=h^{2}$
$\therefore h=\sqrt{2} r$
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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,