Question

Find the volume common to two spheres, each with radius r, if the distance between their centers is r/2.

(given information: A cap of a sphere with radius r = 5 and height h = 2.)

Answer 1

To find the volume common to two spheres with radius r, subtract the volumes of the caps created by slicing the spheres along a plane parallel to their centers.

Step-by-step explanation:

To find the volume common to two spheres with radius r, we can imagine slicing the two spheres along a plane parallel to their centers. This would create two caps, each with a height of h=r/2. We can use the formula for the volume of a cap of a sphere, which is V=(pi*h^2*(3r-h))/3. Since the height of the cap is r/2, the volume of each cap is V=(pi*(r/2)^2*(3r-r/2))/3. To find the volume common to both spheres, we can subtract the sum of the volumes of the two caps from the volume of each sphere, which is V=(4/3)*(pi*r^3).