The inside of a bowl is part of a sphere. When water is put into the bowl to a depth d, the water surface becomes a circle of radius 2d. What is the radius of the sphere?

Let R be the radius of the sphere. The water surface is a chord-circle at depth d from the top of the spherical cap. If the depth of water is d and the radius of the water surface is 2d, by the geometry of spherical caps: (2d)² + (R − d)² = R². So 4d² + R² − 2Rd + d² = R², giving 5d² = 2Rd, so R = 5d/2 = 2.5d.