Posted by Marty on Monday, March 9, 2009 at 8:01pm.
Find the volume of the cone of maximum value that can be inscribed in a sphere of radius 10cm.
Calculus - Reiny, Monday, March 9, 2009 at 8:16pm
Make a diagram.
let the radius of the cone be x,
let the distance from the base of the cone to the centre of the sphere by y,
then the height of the cone is y+10
draw a line from the centre of the sphere to the end of the radius of the cone.
You now have a right-angled triangle where
x^2 + y^2 = 100
and x^2 = 100 - y^2
volume of a cone = (1/3)pi(r^2)h
= (pi/3)(100 - y^2)(y+10)
= (pi/3(100y + 1000 - y^3 - 10y^2)
d(volume)/dy = pi/3(100 - 3y^2 - 20y)
= 0 for a max/min of volume
so 3y^2 + 20y - 100 = 0
(3y-10)(y+10) = 0
then y = 10/3 or y = -10 which would be silly
so y = 10/3
sub that back in x^2 - 100 - y^2
and then into the volume of the cone
Answer This Question
More Related Questions
- Calculus - A cone is inscribed in a sphere of radius a, centred at the origin. ...
- calculus - Find the radius of the base and altitude of a right circular cone of ...
- Calculus - A right circular cone is inscribed in a sphere of radius r. Find the...
- Calculus - A right circular cone is inscribed in a sphere of radius r. Find the ...
- Math - A sphere is inscribed in a right cone. The slant height of the cone = the...
- Geometry 10th - A cone is inscribed in a sphere with a radius of 5 centimeters....
- Calculus - Find the largest volume V of the circular cone that can be inscribed ...
- Math - (a) A sphere of radius 4 in is inscribed in a cone with radius 6 in. How ...
- math - A right circular cone is inscribed inside a sphere. The right circular ...
- I need help on a geometry cone/frustrum problem - A sphere with radius 3 is ...