A cylindrical bucket, $ 32 \mathrm{~cm} $ high and $ 18 \mathrm{~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 \mathrm{~cm} $, find the radius and slant height of the heap.
Given:
A cylindrical bucket, \( 32 \mathrm{~cm} \) high and with radius of base \( 18 \mathrm{~cm} \), is filled with sand.
This bucket is emptied out on the ground and a conical heap of sand is formed.
The height of the conical heap is \( 24 \mathrm{~cm} \)
To do:
We have to find the radius and slant height of the heap.
Solution:
Radius of the cylindrical bucket $r=18 \mathrm{~cm}$
Height of the cylindrical bucket $h=32 \mathrm{~cm}$
This implies,
Volume of the sand in the bucket $=\pi r^{2} h$
$=\pi(18)^{2} \times 32$
$=\pi \times 324 \times 32$
$=10368 \pi \mathrm{cm}^{3}$
Height of the conical heap $H=24 \mathrm{~cm}$ Let the radius of the conical heap be $R$.
This implies,
Volume of the conical heap $=\frac{1}{3} \pi R^{2} H$
$\Rightarrow 10368 \pi=\frac{1}{3} \times \pi R^{2} \times 24$
$\Rightarrow R^{2}=\frac{10368 \pi \times 3}{\pi \times 24}$
$=1296$
$=(36)^{2}$
$\Rightarrow r=36 \mathrm{~cm}$
Therefore,
Radius of the conical heap $=36 \mathrm{~cm}$
Slant height of the heap $l=\sqrt{R^{2}+H^{2}}$
$=\sqrt{(36)^{2}+(24)^{2}}$
$=\sqrt{1296+576}$
$=\sqrt{1872}$
$=\sqrt{144\times13}$
$=12\sqrt{13}\ cm$
The radius and slant height of the heap are $36\ cm$ and $12\sqrt{13}\ cm$ respectively.
Related Articles
- A cylindrical bucket of height \( 32 \mathrm{~cm} \) and base radius \( 18 \mathrm{~cm} \) 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 \( 24 \mathrm{~cm} \), find the radius and slant height of the heap.
- A cylindrical bucket, \( 32 \mathrm{~cm} \) high and with radius of base \( 18 \mathrm{~cm} \), is filled with sand. This bucket is emptied out on the ground and a conical heap of sand is formed. If the height of the conical heap is \( 24 \mathrm{~cm} \), find the radius and slant height of the heap.
- 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.
- The \( \frac{3}{4} \) th part of a conical vessel of internal radius \( 5 \mathrm{~cm} \) and height \( 24 \mathrm{~cm} \) is full of water. The water is emptied into a cylindrical vessel with internal radius \( 10 \mathrm{~cm} \). Find the height of water in cylindrical vessel.
- Find the capacity in litres of a conical vessel with(i) radius \( 7 \mathrm{~cm} \), slant height \( 25 \mathrm{~cm} \)(ii) height \( 12 \mathrm{~cm} \), slant height \( 13 \mathrm{~cm} \).
- If the radii of the circular ends of a conical bucket which is \( 45 \mathrm{~cm} \) high be \( 28 \mathrm{~cm} \) and \( 7 \mathrm{~cm} \), find the capacity of the bucket. ( Use \( \pi=22 / 7 \) ).
- A conical vessel, with base radius 5 cm and height 24 cm, is full of water. This water is emptied into a cylindrical vessel of base radius 10 cm. Find the height to which the water will rise in the cylindrical vessel.
- If the radii of the circular ends of a bucket \( 24 \mathrm{~cm} \) high are \( 5 \mathrm{~cm} \) and \( 15 \mathrm{~cm} \) respectively, find the surface area of the bucket.
- A conical hole is drilled in a circular cylinder of height \( 12 \mathrm{~cm} \) and base radius \( 5 \mathrm{~cm} \). The height and the base radius of the cone are also the same. Find the whole surface and volume of the remaining cylinder.
- A hollow sphere of internal and external radii \( 2 \mathrm{~cm} \) and \( 4 \mathrm{~cm} \) respectively is melted into a cone of base radius \( 4 \mathrm{~cm} \). Find the height and slant height of the cone.
- The $\frac{3}{4}$th part of a conical vessel of internal radius $5\ cm$ and height $24\ cm$ is full of water. The water is emptied into a cylindrical vessel with internal radius $10\ cm$. Find the height of water in cylindrical vessel.
- Find the volume of the right circular cone with(i) radius \( 6 \mathrm{~cm} \), height \( 7 \mathrm{~cm} \)(ii) radius \( 3.5 \mathrm{~cm} \), height \( 12 \mathrm{~cm} \).
- The height of a cone is \( 15 \mathrm{~cm} \). If its volume is \( 1570 \mathrm{~cm}^{3} \), find the radius of the base. (Use \( \pi=3.14 \) ).
- A copper sphere of radius \( 3 \mathrm{~cm} \) is melted and recast into a right circular cone of height 3 \( \mathrm{cm} \). Find the radius of the base of the cone.
- A bucket made of aluminium sheet is of height \( 20 \mathrm{~cm} \) and its upper and lower ends are of radius \( 25 \mathrm{~cm} \) and \( 10 \mathrm{~cm} \) respectively. Find the cost of making the bucket if the aluminium sheet costs \( ₹ 70 \) per \( 100 \mathrm{~cm}^{2} . \) (Use \( \left.\pi=3.14\right) . \)
Kickstart Your Career
Get certified by completing the course
Get Started