q26)Show that the volume of the cone is `pi/3 (1+x-x^2-x^3)` `Thanks!`
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.
Now we use the values of `r` and `h` to calculate the volume of the cone.
Multiply the terms in brackets.
You can find the cone with the maximum volume by solving equation
Therefore, the cone will have the maximum volume for `x=1/3.`