The lengths of the adjacent sides of right angle of a right - angled triangle are 20 cm and 15cm. What will be the total volume of the two right circular cones when this triangle is rotated with its hypotenuse taken as the axis ?

Explanation

Let the perpendicular of the right -angled triangle ABC be BC=20cm and its base , AB=15cm.
`therefore` hypotenuse , `AC=sqrt((20)^2+(15)^2)cm`.
`=sqrt(400+ 225)cm=sqrt(625)cm=25cm`
Now , perpendicular BO is drawn from B to the hypotenuse AC.
`therefore` the area of the `Delta ABC=(1)/(2)xxABxxBC=(1)/(2)xxACxxBC=(1)/(2)xxACxxBO`
`rArr ACxxBO=ABxxBCrArr 25xxBO=15xx20`
`rArr BO=(15xx20)/(25)rArr BO=12`
So, the radius of two cones produced by rotating the triangle with the hypotenuse AC as the axis will be 12 cm .
Also , the height of the two cones will AO and CO.
So, the total volume of the two cones `=((1)/(3)pixx12^2xxAO+(1)/(3)pixx12^2xxCO)c c`.
`=(1)/(3)pi xx(12)^2xx(AO+BC) c c =(1)/(3)pi xx 144 xxAC c c`
`=(1)/(3)xx144xx25pi c c =1200 pi c c`.
Hence the total volume of the two produced cones `=1200 pi c c`.