Show that the semi vertical angle of a right circular cone of given total surface area and maximum
volume is sin^-1 (1/3)

Question

Show that the semi vertical angle of a right circular cone of given total surface area and maximum volume is sin^-1 (1/3)

SHAIK AASIF AHAMED · Answer

If the height is h and the base radius is r we have
V = ⅓πr²h … (i)
S = πr√(r²+h²) + πr² … (ii) … (S is a constant)
From (ii) (S/π−r²)² = r⁴+ r²h² → S²/π²−2Sr²/π = r²h² … (iii)
Anticipating use in (i) this gives h²r⁴ = S²r²/π²−2Sr⁴/π
Using (i) V² = (π²/9)( S²r²/π²−2Sr⁴/π ) = (S/9)( Sr²−2πr⁴ )
V is greatest when V² is greatest so we just set derivative to zero
(S/9)( 2Sr−8πr³ ) = 0 → r² = S/(4π)
Substituting in (iii) leads to h² = 2S/π
∴ tan(θ) = r/h = 1/√8
From a right-angled triangle of sides 1, √8 and 3, sin(θ) = ⅓

θ = $\sin^{-1}(1/3)$

Thanks & Regards

Jitender Singh

IIT Delhi

askIITians Faculty

Thank you for registering.

Register Now and Win Upto 25% Scholorship for a Full Academic Year !

Enter Your Details

Registration done!

Guest

Show that the semi vertical angle of a right circular cone of given total surface area and maximum volume is sin^-1 (1/3)

Show that the semi vertical angle of a right circular cone of given total surface area and maximum
volume is sin^-1 (1/3)

1 Answers

Think You Can Provide A Better Answer ?

Other Related Questions on Algebra

ASK QUESTION

Answer and earn gift vouchers

View courses by askIITians

Design classes One-on-One in your own way with Top IITians/Medical Professionals

Complete Self Study Package designed by Industry Leading Experts

Live 1-1 coding classes to unleash the Creator in your Child

a Complete All-in-One Study package Fully Loaded inside a Tablet!

Register Now & Win Upto 25% Scholorship

Thank you for registering.

Register Now and Win Upto 25% Scholorship for a Full Academic Year !

Enter Your Details

Registration done!

Guest

IIT JEE Courses

One Year IIT Programme

Two Year IIT Programme

Crash Course

NEET Courses

One Year NEET Programme

Two Year NEET Programme

One to One

School Board

Live Online classes

Class 11 & 12

Class 9 & 10

Class 6, 7 & 8

Test Series

JEE test series

NEET test series

C.B.S.E test series

Complete Self Study Packages

FULL COURSE

For class 12th

For class 11th

Show that the semi vertical angle of a right circular cone of given total surface area and maximum volume is sin^-1 (1/3)

Show that the semi vertical angle of a right circular cone of given total surface area and maximum volume is sin^-1 (1/3)

1 Answers

Think You Can Provide A Better Answer ?

Other Related Questions on Algebra

ASK QUESTION

Answer and earn gift vouchers

View courses by askIITians

Design classes One-on-One in your own way with Top IITians/Medical Professionals

Complete Self Study Package designed by Industry Leading Experts

Live 1-1 coding classes to unleash the Creator in your Child

a Complete All-in-One Study package Fully Loaded inside a Tablet!

Register Now & Win Upto 25% Scholorship

Show that the semi vertical angle of a right circular cone of given total surface area and maximum
volume is sin^-1 (1/3)