A colth having an area of `165 m^(2)`is shapped into the form of a conical tent of radius 5 m.
(i) How many students can sit in the tent, if a student on an average occupies `(5)/(7)m^(2)` on the ground ?
(ii) Find the volume of the cone.

Explanation

(i) Given, radius of the base of conical tent = 5 m
and area needs to sit a student on the ground `= (5)/(7)m^(2)`
`therefore` Area of the base of a conical tent `= pir^(2)`.
`= (22)/(7) xx 5 xx 5m^(2)`
Now, number of students `= ("Area of the bases of a conical tent")/("Area needs to sit a student on the groud")`
`= ((22xx5xx5)/(7))/(5//7) = (22)/(7) xx 5 xx 5xx(7)/(5) = 110`
Hence, 110 students can sit in the conical tent .
(ii) Given, area of the form a conical tent `= 165m^(2)`
Radius of the base of a conical tent , r = 5 m
Curved surface area of the = Area of cloth to from a conical tent
`rArr" "pirl = 165`
`rArr" "(22)/(7)xx (5) xx l = 165`
`therefore" " l = (165 xx 7)/(22 xx5) = (33xx7)/(22) = 10.5m`
Now, height of a conical tent = `sqrt(l^(2) - r^(2)) = sqrt((10.5)^(5) - (5)^(2))`
` = sqrt(110.25 - 25 ) = sqrt(8528) = 9.23m`
Volume of a cone (conical tent) `=(1)/(3) pir^(2)h = (1)/(3) xx (22)/(7) xx 5 xx 5 xx 923`
` = (1)/(3) x (1550xx 923)/(7) = (50765)/(7xx3) = 241.7m^(3)`
Hence, the volume of the (conical tent ) is `241.7m^(3)`