Question

A well of diameter 4m is dug 14m deep. The earth taken out is spread evenly all around the well to form a 40cm high embankment. Find the width of the embankment.

Answer 1

Hint: According to the given in the question that the earth taken out is spread evenly all around the well to form a 40cm high embankment from a well of earth taken out is spread evenly all around the well to form a 40cm high embankment. Hence, to find the width of the embankment first of all we have to find the volume of the well as given that diameter of the well 4m and depth of the well is 14m but to find the volume we need to recognize the shape, as we know that a well in a shape of cylinder. Hence, we have to find the volume of the cylinder with the help of the formula given below:

Formula used: Volume of the cylinder = $\pi {r^2}h$…………………………..(1)
Where, r is the radius of the cylinder and h is the height of the cylinder

Complete step-by-step answer:
Given,
The diameter of the well = 4m and,
Depth of the well is = 14m
Height of the embankment form is = 40cm.
Step 1: As given in the question, that the earth taken out is spread evenly all around the well to form a 40cm high embankment from a well of earth taken out is spread evenly all around the well to form a 40cm high embankment. So we have to find the volume of the well but to find the volume we need to recognize the shape, as we know that a well is in the shape of a cylinder. So first of all we have to find the radius of the well.
Diameter of the well = 4m
As we know that radius is equal to half of the diameter.
Radius of well $ = \dfrac{4}{2}m$
Radius of well = 2m
Step 2: Now, we can find the volume of the well with the help of the formula (1) as mentioned in the solution hint.
As, given in the question that the depth of the well is 14m and the radius is 2m.
Hence, on substituting all the values,
Volume of the well $ = \pi {(2)^2} \times 14$
As we know that value of $\pi = \dfrac{{22}}{7}$
On substituting the value of $\pi $in the volume of well,
Volume of the well$ = \dfrac{{22}}{7} \times 4 \times 14$
On solving the obtained equation,
Volume of the well$ = 22 \times 8$
Volume of the well\[ = 176{m^3}\]
Step 3: Now, we will find the volume of the embankment. To find the volume first of all we have to recognize the shape of the earth taken out is spread evenly all around the well to form a 40cm high embankment. As we know that the shape of the embankment is in the shape of a cylinder. But first of all to find the volume we have to convert the height of embankment in m.
As we know that,
1 m = 100 cm,
Hence, height of the embankment $ = \dfrac{{40}}{{100}}m$
Height of the embankment$ = 0.4m$
Hence, Volume of embankment $ = \pi \times {(r)^2} \times 0.4$
As we know that $\pi = \dfrac{{22}}{7}$
Volume of embankment$\dfrac{{22}}{7} \times {(r)^2} \times 0.4{m^3}$
Step 4: Now, as given in the question that earth is dug out to form an embankment so, we can say that the volume of well dug is equal to the volume of embankment form. Hence, on comparing the volume of well and the volume of embankment we can find the width of the embankment.
$\Rightarrow$$\dfrac{{22}}{7} \times {(r)^2} \times 0.4 = 176$
On solving the expression we can obtain the width of the well,
$
\Rightarrow {r^2} = \dfrac{{176 \times 7}}{{22}} \\
\Rightarrow {r^2} = 56 \\
\Rightarrow r = \sqrt {56} \\
\Rightarrow r = 7.48m \\
 $
Hence, width of the well = 7.48m

Final solution: the width of the embankment when a well of diameter 4m is dug 14m deep. The earth taken out is spread evenly all around the well to form a 40cm high embankment is 7.48m.

Note: To make the calculation we can use $\pi $ as it is with-out placing its value as $\dfrac{{22}}{7}$ or $3.14$.
The volume of the mud to form an embankment is equal to the volume of the volume of the embankment.
It is necessary to convert the units as the dimensions given in the question.
As we know every well and embankment is in the shape of a cylinder so to find the volume of well and embankment we have to find the volume of the cylinder.