Question

A cylindrical vessel 32 cm high and 18 cm as the 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, the radius of its base is
(a)12 cm
(b)24 cm
(c)36 cm
(c)48 cm

Answer 1

Answer:

The radius of heap is 36 cm.

Among the given options option (c) 36 cm is the correct answer.

Step-by-step explanation:

Given :  

Height of a cylindrical bucket , H = 32 cm  

Radius of cylindrical bucket , R = 18 cm

Height of the conical heap of sand , h = 24 cm

 

Let the radius and slant height of the heap of sand be ‘r’  & ‘ l’.

 

Here, the sand filled in cylindrical bucket from a conical heap of sand on the ground. So volume of cylindrical bucket will be equal to the volume of conical heap.

 

Volume of cylindrical bucket = Volume of conical heap of sand  

πR²H = 1/3 πr²h  

R²H = 1/3 r²h  

18² × 32 = ⅓ × r² × 24

18 × 18 × 32 = 8r²  

r² = (18 × 18 × 32)/8

r² = 18 × 18 × 4

r² = 1296  

r = √1296

r = 36 cm

Radius of the heap of sand  = 36 cm

Hence the radius of heap is 36 cm.

HOPE THIS ANSWER WILL HELP YOU…

Answer 2

As the sand present in the cylindrical bucket is emptied into the conical heap so both of their volume will be equal

As we know that

Volume of cylindrical vessel

Given,

radius = 18 cm

height = 32 cm

Volume of the bucket = Π × r² × h

= Π (18)² 32

volume of conical heap

Given ,

height = 24 cm

Let us assume its radius be "r"

Slant height = l cm

$\boxed{\bold{\mathsf{volume\: of\: conical\: heap = \frac{1}{3} \pi {r}^{2} h}}}$

$\boxed{\bold{\mathsf{= \frac{1}{3} \pi {r}^{2} 24} }}$

$\boxed{\bold{\mathsf{8 \pi r^{2} }}}$

Now , just equate both of thier volumes

Volume of cylindrical bucket/vessel = volume of conical heap

Π (18)² 32 = 8 Π r²

${r}^{2} = \frac{\pi \times {18}^{2} \times 32}{8\pi} \\ \\ {r}^{2} = \frac{ 18 \times 18 \times 32}{8} \\ \\ {r}^{2} = 18 \times 18 \times 4 \\ \\ {r}^{2} ={ (18 \times 2)}^{2} \\ \\ r = 18 \times 2 \\ \\ r = 36cm$

$\rule{200}{2}$

$\boxed{\bold{\mathsf{Formula's \:used}}}$:—

$\underline{\underline{\bold{\mathtt{Volume \:of \:Cylinder = \pi {r}^{2} h }}}}$

$\boxed{\bold{\mathsf{Volume\:of\:cone= \frac{1}{3}\times \pi {r}^{2}h }}}$