F.3 Maths

A metal solid right circular cone of base radius 4cm and height 12cm is put into a ollow cylinder with the same base radius and height as the cone. Then the cylinder is completely filled with water.

The volume of water in the cylinder is 128Π.

Later, some water is pumped out of the cylinder such that the depth of water becomes 9cm. Find the volume of the water pumped out.
首先﹐直角圓錐體的體積是(1/3)Π4^2*12=64Π cm^3
圓桂體的體積是Π(4^2)(12)=192Π cm^3
放個直角圓錐體入圓桂體後
圓桂體的體積-直角圓錐體的體積=128Π cm^3
說明最初水是汪滿整個圓柱體的剩餘空間
水的高度原本12cm
現在抽水抽到水的高度變成9 cm
所以抽水抽了3cm 高
這3cm高的水所佔的空間
=該3cm高的圓桂體的體積-該3cm高的直角圓錐體的體積
=Π(4^2)(3)-(1/3)Π(1^2)(3)
=48Π-Π
=47Π cm^3
換句話說﹐
抽走的水所佔的體積是47Π cm^3
按:為甚麼該3cm高的直角圓錐體的半徑是1 cm?
因為由相似三角形
高是12 cm﹐底4cm
高是3cm, 底便是1 cm了