A cylinder has radius 2cm and height 9cm. A cone has radius r cm and height h cm. The ratio r : h is 1 : 4. The volume of the cone is equal to the volume of the cylinder. Work out the value of r.

Firstly, we will find the volume of the cylinder using the formula V_cylinder =π r²h . Inputting the numbers given in the question gives V_cylinder = 36π.Given that V_cylinder = V_cone, and using the formula V_cone = 1/3 π r²h, we get the expression36π = 1/3 π r²h. The ratio r:h is 1:4, which gives h = 4r. Subbing this into the above expression and rearranging, we getr³= 27. Taking the cube root of both sides gives us r =3cm.