Question

A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap

Answer 1

36 cm

43.27 cm

Explanation:

`r_1` = Radius of bucket = 18 cm

`h_1` = Height of bucket = 32 cm

`r_2` = Radius of cone

`h_2` = Height of cone = 24 cm

The volume of the cylindrical bucket and the conical heap of sand is equal so

`\pi r_1^2h_1=(1)/(3)\pi r_2^2h_2\\\Rightarrow r_2=\sqrt{(3r_1^2h_1)/(h_2)}\\\Rightarrow r_2=\sqrt{(3* 18^2* 32)/(24)}\\\Rightarrow r_2=36\ \text{cm}`

The radius of the heap is 36 cm.

Slant height is given by

`l=√(r_2^2+h_2^2)\\\Rightarrow l=√(36^2+24^2)\\\Rightarrow l=43.27\ \text{cm}`

The slant height of the heap is 43.27 cm.