q26)Show that the volume of the cone is `pi/3 (1+x-x^2-x^3)` `Thanks!`

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tiburtius | High School Teacher | (Level 2) Educator

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Volume of a cone is given by formula `V=pi/3 r^2h` , where `r` is the radius of the base and `h` is the height of the cone.

Since this is a unit sphere (sphere with radius of length 1) the distance `|OA|=1` (because that is the radius.) Therefore the height of the cone is `h=1+x.` From the right angle triangle in the image below we can calculate the radius of the base. Hypotenuse is 1 because it is the radius of the sphere.

`r^2=1-x^2`

Now we use the values of `r` and `h` to calculate the volume of the cone.

`V=pi/3(1-x^2)(1+x)`

Multiply the terms in brackets.

`V=pi/3(1+x-x^2-x^3)`

You can find the cone with the maximum volume by solving equation

`V'=0`

`pi/3(1-2x-3x^2)=0`

`1-2x-3x^2=0`

`x_1=-1, x_2=1/3`

Therefore, the cone will have the maximum volume for `x=1/3.`

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