Question

. A cylindrical bucket 32 cm high and with radius ofbase 18 cm
is filled with sand. The 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

Given:-

• Height of the cylindrical bucket = 32 cm
• Radius of the base = 18 cm
• Height of the conical sand heap = 24 cm

To Find:-

• Radius of the conical heap
• Slant height of the conical heap

Assumption:-

Let the radius of the conical heap be r and length be l

Solution:-

Firstly we'll find the Volume of the cylindrical bucket,

So,

Height = 32 cm

Radius of base = 18 cm

We know,

$\sf{Volume\:of\:cylinder = \pi r^2 h\:\:cu.units}$

Therefore,

$\sf{Volume\:of\:Cylinder = \pi \times (18)^2 \times 32}$

= $\sf{Volume = \pi 10368\:cm^3}$

$\sf{\implies Volume = 10368\pi\:\:cm^3}$

Now,

We'll find the volume of the conical sand heap,

Height = 24 cm

Radius = r

We know,

$\sf{Volume\:of\:cone = \dfrac{1}{3}\pi r^2 h}$

Therefore,

$\sf{Volume\:of\:heap = \dfrac{1}{3}\times \pi \times r^2 \times 24}$

= $\sf{Volume = 8\pi r^2\:cm^3}$

Now,

$\because$ The cylindrical bucket was emptied on a ground,

Volume of heap = Volume of bucket

Therefore,

$\sf{10368\pi = 8\pi r^2}$

= $\sf{\dfrac{10368\pi}{8\pi} = r^2}$

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

=> $\sf{r = \sqrt{1296}}$

= $\sf{r = 36\:cm}$

Therefore the radius of the conical heap is 36 cm.

Now,

Finding the slant height of the conical heap,

Since we have,

radius (base) = 36 cm

height = 24 cm

We can apply Pythagoras theorem to find the slant height of the heap.

We know slant height of a cone = Hypotenuse of the triangle

Therefore, we need to find hypotenuse (l) ,

According to the Pythagoras theorem,

$\sf{(Hypotenuse)^2 = (base)^2 + (height)^2}$

= $\sf{l = \sqrt{(36)^2 + (24)^2}}$

= $\sf{l = \sqrt{1296 + 576}}$

= $\sf{l = \sqrt{1872}}$

= $\sf{l = 43.3\:cm}$

Therefore slant height of the heap is 43.3 cm.

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Note!!

Refer to the attachment for the diagram of finding slant height of the heap.

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