Question

A cylindrical bucket, 32 cm high and 18 cm of radius of the base, 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

The height and radius of the cylindrical bucket are h = 32 cm and r = 18 cm respectively. Therefore, the volume of the cylindrical bucket is 

V = `pir^2 h`

 =`22/7 xx (18)^2 xx 32`

The bucket is full of sand and is emptied in the ground to form a conical heap of sand of height  h₁ = 24 cm. Let, the radius and slant height of the conical heap be  r₁ cm and  l₁ cm respectively. Then, we have

`l_1^2 = r_1^2 + h_1^2`

⇒`r_1^2 = l_1^2 - h_1^2`

⇒`r_1^2 = l_1^2 - (24)^2`

The volume of the conical heap is

`V_1 = 1/3 pir_1^2h_1`

     `=1/3xx22/7 xxr_1^2 xx24`

      `=22/7 xx r_1^2 xx8` 

Since, the volume of the cylindrical bucket and conical hear are same, we have

V₁ = V 

⇒`22/7 xxr_1^2 xx 8 =22/7 xx(18)^2 xx32`

⇒                  `r_1^2 = (18)^2 xx 4`

⇒                  `r_1 = 18 xx 2`

⇒                  `r_1 = 36`

Then, we have

         `l_1^2 = r_1^2 + h_1^2`

⇒     `l_1^2 =(36)^2 + (24)^2`

⇒      l₁ = 43.27

Therefore, the radius and the slant height of the conical heap are 36 cm and 43.27 cm respectively.