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

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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,

`(dV)/(d theta)=0`

`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)

Or, `cos^2theta-2sin^2theta=0`

`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

`V_(max)=2piR^3cos^2thetasintheta`

`=2piR^3(sqrt2sintheta)^2*sintheta`

`=2piR^3*2sin^3theta`

`=4pi*8^3*1/(sqrt3)^3`

`=4pi*8^3*1/(3)^(3/2)`

`=1238.2 cm^3`

**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**.

**Sources:**

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