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A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment

Solution:

Given that, the depth of the well is 14 m and the diameter is 3 m. The width of the circular ring of the embankment is 4 m.

A figure is drawn below to visualize the shapes.

A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment.

From the above figure, we can observe that the shape of the well will be cylindrical, and earth evenly spread out to form an embankment around the well in a circular ring will be cylindrical in shape (Hollow cylinder) having outer and inner radius.

Volume of the earth taken out from well = Volume of the earth used to form the embankment

Hence, Volume of the cylindrical well = Volume of the hollow cylindrical embankment

Let us find the volume of the hollow cylindrical embankment by subtracting volume of inner cylinder from volume of the outer cylinder.

Volume of the cylinder = πr²h where r and h are the radius and height of the cylinder respectively.

Depth of the cylindrical well, = h₁ = 14 m

Radius of the cylindrical well, = r = 3/2 m = 1.5 m

Width of embankment = 4 m

Inner radius of the embankment, r = 3/2 m = 1.5 m

Outer radius of the embankment, R = Inner radius + Width

R = 1.5 m + 4 m

= 5.5 m

Let the height of embankment be h

Volume of the cylindrical well = Volume of the hollow cylindrical embankment 

πr²h₁ = πR²h - πr²h

πr²h₁ = πh (R² - r²)

r²h₁ = h (R - r )(R + r)

h = [(r²h₁)/(R - r)(R + r)]

h = [((1.5 m)² × 14 m)/(5.5 m - 1.5 m)(5.5 m + 1.5 m)]

= (2.25 m² × 14 m)/(4m × 7 m)

= 1.125 m

Therefore, the height of the embankment will be 1.125 m.

☛ Check: NCERT Solutions Class 10 Maths Chapter 13

Video Solution:

A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment

NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.3 Question 4 

Summary:

If a well of diameter 3 m is dug 14 m deep and the earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment, the height of the embankment will be 1.125 m.

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