Thank you for registering.

One of our academic counsellors will contact you within 1 working day.

Please check your email for login details.
MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping

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)

Grade:Upto college level

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
7 years ago
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

Think You Can Provide A Better Answer ?

Provide a better Answer & Earn Cool Goodies See our forum point policy

ASK QUESTION

Get your questions answered by the expert for free