Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin inverse 1 by 3

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

Answers (1)

best_answer


Let r, l, and h are the radius, slant height and height  of cone respectively
Now,
r = l\sin a \ and \ h =l\cos a
Now, we know that
The surface area of the cone (A) = \pi r (r+l)
A= \pi l\sin a l(\sin a+1)\\ \\ l^2 = \frac{A}{\pi \sin a(\sin a+1)}\\ \\ l = \sqrt{\frac{A}{\pi \sin a(\sin a+1)}}
Now, Volume of cone(V) =

 \frac{1}{3}\pi r^2h = \frac{1}{3}\pi l^3 \sin^2 a\cos a= \frac{\pi}{3}.\left ( \frac{A}{\pi\sin a(\sin a+1)} \right )^\frac{3}{2}.\sin^2 a\cos a
On differentiate it w.r.t to a and after that
\frac{dV}{da}= 0
we will get 
a = \sin^{-1}\frac{1}{3}
Now, at  a = \sin^{-1}\frac{1}{3}
\frac{d^2V}{da^2}<0
Hence, we can say that a = \sin^{-1}\frac{1}{3} is the point if maxima 
Hence proved.

Posted by

Gautam harsolia

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

PM Modi's Pariksha Pe Charcha 2025 to be held next month, registrations open till January 14
anam-khan

CBSE Board Exams 2025: Online portal for availing exemptions by CWSN students opens
anam-khan

Board Exams: States agree to equivalence; no question paper ‘jumbling’ from next year, says PARAKH CEO

Latest Question