A cylinder of cheese is to be removed from a spherical piece of cheese with a radius of 8 cm what's the maximum volume of the cylinder piece of cheese
Draw a circle on the plane with it's centre at the origin and a rectangle in it representing a 2D slice of the cylinder.
The radius of the circle, R = 8 cm points to one corner of the rectangle.
The radius of the cylinder (i.e. half the width of rectangle) is `r = Rcostheta`
the height of the cylinder is `h = 2Rsintheta`
Volume of the cylinder `V = pir^2h`
Putting the values of r and h we get:
`V = pi(Rcostheta)^2(2Rsintheta) = 2piR^3cos^2thetasintheta`
For extremum (maximum or minimum) value of V,
`rArr 2piR^3(cos^2theta*costheta + sintheta*2costheta*(-sintheta)) = 0`
`rArr 2piR^3(cos^3theta - 2sin^2thetacostheta)=0`
`rArr 2piR^3costheta(cos^2theta - 2sin^2theta)=0`
Therefore, Either, `costheta=0` , That corresponds to V=0 (minimum value)
`rArr cos^2theta = 2sin^2theta`
`rArr tan^2theta = 1/2`
Or,`tantheta = 1/sqrt2`
Forming a right triangle with this tan value yields
`sintheta = 1/sqrt3 and costheta = sqrt2sintheta`
Therefore, the maximum volume of the cylindrical piece of cheese
So, the maximum volume of the cylindrical piece of cheese that can be cut out from the spherical cheese ball is 1238.2 cm^3.