calculus
posted by anu .
A right triangle of hypotenuse L is rotated about one of its legs to generate a right circular cone
find the largest volume that such a cone could occupy

Let cone height be h, which is one of the two legs. The base radius will be
R = sqrt(L^2  h^2)
Volume = (1/3)pi*R^2*h
= pi*(L^2  h^2)*h/3
Volume is a maximum when dV/dh = 0
dV/dh = pi*L^2  pi*3*h^2 = 0
L^2 = 3 h^2
Max volume = (1/3) pi*(L^2L^2/3)*sqrt(1/3)L
= 2*pi*L^3/(9*sqrt3)
Respond to this Question
Similar Questions

calculus
A right triangle of hypotenuse L is rotated about one of its legs to generate a right circular cone find the largest volume that such a cone could occupy 
math
a right triangle whose hypotenuse is h mm long is rotated about one of its legs to generate a circular cone.find the radius and altitude of the cone of greatest volume. 
Calculus
A right circular cone is inscribed in a sphere of radius r. Find the dimensions of the cone that maximize the volume of the cone. 
Calculus
A right circular cone is inscribed in a sphere of radius r. Find the dimensions of the cone that maximize the volume of the cone. 
calculus
A right triangle with hypotenuse 5 inches is revolved around one of its legs to generate a right circular cone. Find the radius and height of the greatest volume cone. 
Calculus
Given a right circular cone, you put an upsidedown cone inside it so that its vertex is at the center of the base of the larger cone, and its base is parallel to the base of the larger cone. If you choose the upsidedown cone to have … 
Calculus
Show that a rightcircular cylinder of greatest volume that can be inscribed in a rightcircular cone that has a volume that is 4/9 the volume of the cone. Please, help out, this is for a study guide for a quiz. :) Thank you. 
calculus
Given a right circular cone, you put an upsidedown cone inside it so that its vertex is at the center of the base of the larger cone, and its base is parallel to the base of the larger cone. If you choose the upsidedown cone to have … 
Calculus: Optimization
I have no idea how to approach this problem, if someone knows just how to relate h, r with H,R, that would be extremely helpful and I can workout the rest! Thank you in advance. Given a right circular cone, you put an upsidedown cone … 
Maths
A right circular cone is circumscribed about a sphere of radius a .If h is the distance from the centre of the sphere to the vertex of the cone. Show that the volume of the cone is 1/3pi a^2(a+h)^2/(ha). Find the angle of a cone when …