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

Answer:

The radius and slant height of heap are 36 cm & 43.2 cm.

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

Slant height of the conical heap of sand, l = √(h² + r²  

l = √24² + 36² = √(576 + 1296) = √1872

l = √144 × 13 = 12√13  

l = 12√13 cm

l = 12 × 3.6 = 43.2 cm

slant height of the conical heap of sand, l = 43.2 cm

Hence the radius and slant height of heap are 36 cm & 12√13 cm .

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Answer 2

Step by step explanation :

Let the radius and slant height be r cm and L cm respectively.

Radius of the cylindrical bucket = 18 cm

Height of the cylindrical bucket=32 cm

Height of the conical heap =24 cm

The volume of sand remains the same in the bucket and in the heap.

So, Volume of cylindrical bucket = Volume of conical heap

$\tt{\pi{r}^{2}h=\frac{1}{3}{\pi\:r}^{2}h}$

$\tt\small{\pi \times 18 { }^{2} \times 32 = \frac{1}{3} \pi \times r{}^{2}\times24}$

$\tt{{r}^{2}=\frac{3 \times 18\times 18 \times 32}{24}}$

$\tt{{r}^{2} =1296}$

$\tt{r = 36}$

Now,

Slant height of the conical heap =

$\tt{ \sqrt{r{}^{2}+h {}^{2} }}$

$\tt{ =\sqrt{36{}^{2} + 24 {}^{2} }}$

$\tt{=\sqrt{1296 + 576} }$

$\tt{ =\sqrt{1872}}$

$\tt{ = 43.25}$

Thus, Slant height of the heap is 43.25 cm

Radius of the heap is 36 cm.