Question

Find the largest possible volume of a cone that fits inside a sphere of radius one.

Answer 1

@ganeshie8 @Hero

Answer 2

you get the volume of the sphere which is 4.18879

Answer 3

then you just have to find a volume of a cone smaller than that

Answer 4

https://sketch.io/render/sketch55e116f6c4926.png

Answer 5

Use the above diagram and find an equation to represent the volume of the cone in terms of r. Then you can get the answer :-)

Answer 6

Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3?
Let
R = radius sphere
r = base radius cone
R + h = height cone
V = volume cone
_______

V = (1/3)πr²(R + h)

By the Pythagorean Theorem:
r² = R² - h²

Plug into the formula for volume.
V = (1/3)π(R² - h²)(R + h) = (1/3)π(R³ + R²h - Rh² - h³)

Take the derivative and set equal to zero to find the critical points.

dV/dh = (1/3)π(R² - 2Rh - 3h²) = 0
R² - 2Rh - 3h² = 0
(R - 3h)(R + h) = 0
h = R/3, -R

But h must be positive so:
h = R/3

Calculate the second derivative to determine the nature of the critical points.

d²V/dh² = (π/3)(-2R - 6h) < 0
So this is a relative maximum which we wanted.

Solve for r².
r² = R² - h² = R² - (R/3)² = R²(1 - 1/9) = (8/9)R²

Calculate maximum volume.

V = (π/3)[(8/9)R²](R + R/3) = (8/27)πR³(4/3) = 32πR³/81

For R = 3 maximum volume is:

V = 32π(3³)/81 = 32π(27)/81 = 32π/3

Answer 7

this was the question that came in our examination and this is how i solved it ....and this is similar to your problem only difference is that you gotta put 1 in place 3 (radius):))))))