The bowl has a radius of 10cm. The distance from the height of the water to the top centre of the hemisphere (from where the radius is measured) is cm

I drew a diagram in front of me for this, and you can do it using Pythagoras.

Draw the hemisphere. Draw the level of the water at any distance cm below the top and label the distance. Then draw a radius from the centre top of the hemisphere to the point on the hemisphere that touches the surface of the water. If you have understood me, you should now have a right triangle. The base is , the height is and the hypotenuse is .

I hope you'll forgive me for my dreadful artistic skills, but I have done a diagram here:

Spoiler:

I'll check this in a second, it's sort of a leap of faith on my part, but hopefully you should get the expected result.

Edit: Yeah, I've managed to obtain their solution this way.