Question

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 shape of a circular ring of width 4 m to form an embankment. Find the height of the
embankment​

Answer 1

Given:

• Height of the well, h₁ = 14 m
• Diameter of the well, d = 3 m
• Radius of the well, r₁ = 3/2 m
• Width of embankment = 4 m

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To find:

• Height of the embankment, h₂ ?

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Solution:

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• Outer radius, r₂ = 4 + 3/2 = 11/2 m

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☯ Let height of embankment be h₂.

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$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

Volume of soil dug from well = Volume of earth used to form the embankment

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$:\implies\sf \pi \times {r_1}^2 \times h_1 = \pi \times ({r_2}^2 - {r_1}^2) \times h_2$

$:\implies\sf \cancel{\pi} \times \bigg( \dfrac{3}{2} \bigg)^2 \times 14 = \cancel{\pi} \times \bigg[ \bigg( \dfrac{11}{2}\bigg)^2 - \bigg( \dfrac{3}{2} \bigg)^2 \bigg] \times h_2$

$:\implies\sf \dfrac{9}{4} \times 14 = \bigg[\dfrac{121}{4} - \dfrac{9}{4} \bigg] \times h_2$

$:\implies\sf \dfrac{9}{4} \times 14 = \bigg[\dfrac{121 - 9}{4}\bigg] \times h_2$

$:\implies\sf \dfrac{9}{4} \times 14 = \dfrac{112}{4} \times h_2$

$:\implies\sf h_2 = \dfrac{9}{ \cancel{4}} \times 14 \times \dfrac{ \cancel{4}}{112}$

$:\implies\sf h_2 = 9 \times \cancel{14} \times \dfrac{1}{ \cancel{112}}$

$:\implies\sf h_2 = \dfrac{9}{8}$

$:\implies{\boxed{\sf{\purple{h_2 = 1.125\;m}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Hence,\;The\; height\;of\;embankment\;is\; \bf{1.125\;m}.}}}$

Answer 2

To Find:-

• The height of the embankment ..(?)

Solution:-

Let the well be cylinder A and embankment be cylinder B.

As both well and embankment are in the form of a cylinder.

Now, Let's find the volume of well (Cylinder A)

Given that, Diameter = 3m and height = 14m

So, radius = diameter/2

» 3/2

» 1.5m

We know that,

Volume of Cylinder ( A ) = πr²h

Therefore,

» π × ( 1.5 )² × 14

» π × 2.25 × 14

» 31.5 π m³

Therefore, Volume of well is 31.5 π m³

Now, let's find the volume of embankment ( Cylinder B )

As cylinder B is a hollow cylinder,

Therefore, Inner diameter = diameter of the well = 3m

So, internal radius = 1.5 m

Now, external radius = internal radius + width

» 1.5 + 4

» 5.5 m

Now, volume of Cylinder with internal radius :-

» π ${r_{1}}^{2}$ h

» πh ( 1.5 )²

Volume of Cylinder with external radius = π ${r_{2}}^{2}$ h

» πh ( 5.5 )²

Then, Volume of cylinder B = Volume of cylinder with external radius - Volume of cylinder with internal radius

» πh ( 5.5 )² - πh ( 1.5 )²

» πh { ( 5.5 )² - ( 1.5 )² }

» πh ( 30.25 - 2.25 )

» πh ( 28 )

» 28πh m³

So, now the volume of well = Volume of cylinder

: ⟹ 31.5 π = 28 πh

: ⟹ 28 πh = 31.5 π

: ⟹ h = 31.5 / 28

: ⟹ h = 4.5 / 4

: ⟹ h = 1.125

Required Answer:-

The height of the embankment is 1.125 m.