Question

A circular tank of diameter 2 m is dug and the earth removed is spread uniformly all around the tank to form an embankment 2 m in width and 1.6 m in height. Find the depth of the circular tank.
A. 2.26 m
B. 12.8 m
C. 2.76 m
D. 2.36 m

Answer 1

Hint: In this question it is given that a circular tank of diameter 2 m is dug and the earth removed is spread uniformly all around the tank to form an embankment 2 m in width and 1.6 m in height. We have to find the depth of the circular tank. So to understand it in better way let us draw the diagram,

So to find the depth( height) of the circular tank we need to know the formula of hollow cylinder and the volume of a cylindrical tank,
Therefore, the volume of hollow cylinder,
$$V_{1}=\pi \left( R^{2}-r^{2}\right) H$$.........(1)
Where,
R= radius of the outer radius of the hollow cylinder
r= inner radius of the hollow cylinder
H= height of the hollow cylinder
And the volume of a cylindrical tank,
$$V_{2}=\pi r^{2}h$$..........(2)
Where,
h= height of the cylinder
r= radius of the cylinder

Complete step-by-step answer:
Let us consider the depth of the circular tank= h meter.
Given, the diameter of the inner tank(d)= 2 m
Therefore, the radius(r), i.e, OA=$$\dfrac{d}{2} =\dfrac{2}{2} \ m=1\ m$$, which is also the inner radius of the hollow cylinder(embankment).
Thus, r=1 m
And since the embankment is 2 m width i,e, AB= 2m, then the outer radius(R)= OA+AB= (1+2) m= 3 m .
And the height of the embankment,i.e, the height(H) of the hollow cylinder, H=1.6 m,
Then by the formula (1)
The volume of the embankment,
$$V_{1}=\pi \left( R^{2}-r^{2}\right) H$$
   $$=\pi \left( 3^{2}-1^{2}\right) \times 1.6$$
   $$=\pi \left( 9-1\right) \times 1.6$$
   $$=\pi \times 8\times 1.6$$.........(3)

Now the radius of the inner cylindrical tank, r= 1 m, and the depth = h meter,
Therefore, the volume,
$$V_{2}=\pi r^{2}h$$
   $$=\pi \times 1^{2}\times h$$
   $$=\pi h$$........(4)
Now, Volume of tank = Volume of embankment
$$\therefore V_{2}=V_{1}$$
$$\Rightarrow \pi h=\pi \times 8\times 1.6$$
$$\Rightarrow \pi h=12.8\pi$$
$$\Rightarrow h=\dfrac{12.8\pi }{\pi }$$
$$\Rightarrow h=12.8$$
Therefore, the depth of the tank is 12.8 m.
Hence the correct option is option B.


Note: While solving this type of question you need to know that when you use removed earth to form another shape then the volume of the removed earth is equal to the volume of that created shape also if the removed earth forming a hollow shape(in this question it is a cylindrical tank) then the volume of that shape is also equal to the removed earth, so because of this reason we equate volume of tank with the volume of embankment.