THROUGH A SPHERE OF RADIUS r, A CYLINDRICAL HOLE OF RADIUS 2" IS DRILLED. FIND THE VOLUME OF WHAT REMAINS.?

A figuree showing this will explain well. We need to consider ---

1] Volume of original sphere of radius = r and deduct from this ---

i] volume of cylinder made by hole of radius 2 and ii] volume of spherical caps made at both ends by this hole.

Start with volu. of Sphere = 4/3 pi * r^3 --------------------------------[A]

Second is vol.of cylinder = pi* (2)^2 *h where h= sqrt.(r^2-4)*2

This equals to ---- 8*pi*sqrt(r^2-4)-------------------------------------------[B]

Next is vol. of 2 caps [ on either ends] = 2 * pi*c^2(r-c/3) where c is the height of cap.

Height of cap is given by { r-h/2) } = r-sqrt( r^2-4)

So vol.of both caps = 2*pi*{ r-sqrt(r^2-4)}^2{ r-[r-sqrt(r^2-4)]/3} -----[C]

Now Simplify and calculate [A] - [B] - [C]

Simplification needs some 8 to 10 steps and the expression finally deduces to==

4/3*pi*sqrt(r^2-4)* (r^2-4) which further reduces to final expression

i.e 4/3*pi*[sqrt(r^2-4)]^3---- this is the remaining volume.

Volume of the remaining sphere (holed one) is given by

(4/3)(22/7)R^3 -- volume removed

= (4/3)(22/7)R^3 -- [ volume of a cylinder of radius 2 and height 2 sqrt(R^2 -- 2^2) + volume of two segments of sphere subtending angle 2 arcsin(2/R) at the centre]

= (4/3)(22/7)R^3 -- [ (22/7)*2^3 sqrt(R^2 -- 4) + 2{ (pi/6) (3*2^2 + h^2)h}] where h = R -- sqrt(r^2 -- 2^2)

Start with the volume of a sphere:

V1 = 4/3*(pi)*r^3

Subtract the volume of the cylinder:

V2 = (pi)*r^2*h = (pi)*2*2*h = (pi)*4*h

So the remaining volume is:

V1-V2 = 4/3*(pi)*r^3-(pi)*4*h

This is not exact, because the cylinder assumes a flat circle on each end, yet the ends of the piece you are drilling out curves along the surfaces of the sphere.

radius of sphere = r

radius of cylinder = 2''

height of cylinder = diameter of sphere = 2r

(1) volume of sphere = 4/3 pi r^3

(2) volume of cylinder = pi (2)^2 .2r = 8pi r

so the the volume of remains = 4/3pi r^3 - 8pi r

= 4pi r( r^2/3 -2) ans

Volume of complete sphere = 4*PI*R^3/3

Volume of cylinder = PI*2^2*2R

Volume Remaining = 4*PI*R^3/3 - PI*4*2R

= 4*PI*R*[R^2/3 - 2]

(4/3)*(pi*r*r*r) - (pi*2*2*l)

but we have l =2r as drills through sphere

=> v(remaining) = 4*pi*r*[(r*r)/3 - 2] cubic units

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