Suppose you drill a circular hole with radius r through the center of a sphere with radius R. You remove exactly half the volume of the sphere. The ratio of your radii is:
r/R=
To find the ratio of the radii, we can use the formula for the volume of a sphere:
V = (4/3)πR^3
Since you removed exactly half the volume of the sphere, the new volume will be:
V_new = (1/2) * (4/3)πR^3
Now, let's find the volume of the drilled hole. The volume of a cylinder can be calculated using the formula:
V_cylinder = πr^2h
Since the hole passes through the center of the sphere, the height of the cylinder is equal to 2R. Therefore, the volume of the hole is:
V_hole = πr^2(2R)
Now, let's subtract the volume of the hole from the new volume of the sphere:
V_new - V_hole = (1/2) * (4/3)πR^3 - πr^2(2R)
Simplifying the equation:
(1/2) * (4/3)πR^3 - πr^2(2R) = (2/3)πR^3 - 2πr^2R
Now, let's cancel out the π terms:
(2/3)R^3 - 2r^2R = (2/3)R^3 - 2r^2R
Since the equation is equal on both sides, we can set the coefficients equal to each other:
(2/3) = -2r^2
Now, solving for r/R:
r/R = √(-2/3)
The ratio of r/R is the square root of -2/3, which is an imaginary number. Therefore, the ratio is a complex number.