My son has asked me this question but I am somewhat lost. Can you please guide?
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A vessel has the shape of an inverted cone.The radius of the top is 8 cm and the height is 20 cm. Water is poured in to a height of x cm.Show that if the volume of the water is V cubic cm,then V=(4/75)pi x³.

Thanks & regards,
Ghulam

Ghulam,

The essential concept here is similar triangles.

In the diagram above I have let r be the radius of the circle formed by the water surface. The triangles PQR and PTS are similar and hence

 ^|QR|/_|RP| = ^|TS|/_|SP|  that is  ⁸/₂₀ = ^r/_x 

Hence

r = ²/₅  x

The volume of a cone of radius r and height h is

V = ¹/₃  r² h

and hence the volume of water in your cone is

V = ¹/₃  r² x =  ¹/₃  ( ²/₅  x)² x =  ⁴/₇₅  x³

Penny