The quantity of iron - sheet to make a boy a of right circular conical shape is `75(3)/(7)` sq-m. if the slant height of it 5 m, then calculate the volume of air in the boya and its height. Determine of the expenditure to colour the whole surface of the boya at the rate of rupees 2.80 per square - metres. [The width of the iron - sheet not to be considered while calculating.]

Explanation

The slant height of the boya is 5 metres.
Let the radius of the boya be r metres.
`therefore` the total surface area of the boya ` =(22)/(7) r(5+r)sq-m`
`therefore (22)/(7)r(5+r)=75(3)/(7) or , (22)/(7)r(5+r)=(528)/(7)or , (r+5)=(528xx7)/(7xx22) or , r(r+5)=24`
`or , r^2+5r-24=0 or , r^2+8r-3r-24=0 or , r(r+8)-3(r+8)=0 or , (r+8)(r-3)=0`
`therefore` either `r+8=0, or r-3=0`
`rArr r=-8 rArr r =3`.
Since the radius can never be negative , `therefore r=3`.
Now , if the height of the boya be h m , then `h^2+3^2=5^2or , h^2=25-9 or , h^2=16 or , h=4`.
`therefore ` the height of the boya is 4 metres.
`therefore` the volume of boya `=(10)/(3)xx(22)/(7)xx3^2xx4` cubic - metres `=(264)/(7)`cubic - metres `=37(5)/(7)` cubic- metres.
Hence the height of the boya =4 metres and the volume of air in the boya `=37(5)/(7)` cubic - metres.
Also , at the rate of rupees 2.80 per metre, the cost of colouring the whole surface of the boya ` "rupees" 2.80xx(528)/(7)= "rupees"211.20`.