Question

well of radius 7 metre is dug 8 metre deep. The earth taken out of it is evenly spread all around it to a width of 21 to form an embankment. What is the height of the embankment?​

Answer 1

$\large{\underbrace{\underline{\bf{Required\:Answer\::}}}}$

$\frak{Given}\begin{cases}\sf{Radius\:of\:Well=\bf{7\:m.}}\\\sf{Deapth\:of\:Well=\bf{8\:m.}}\\\sf{Width\:of\:Embankment=\bf{21\:m.}}\end{cases}$

$\bigstar\;\underline{\textit{\textbf{To\:Find\::}}}$

• Height of embankment .

$\bigstar\;\underline{\textit{\textbf{Solution\::}}}$

» We would first find the volume of earth taken out from the well.

» Then we would find the volume of embankment formed.

» Since the earth that is taken out of the well is used to form the embankment so, the volume of the well would be equal to the value of the embankment.

» We would equate the volumes so as to get the value of height of the embankment.

◆ Finding the Volume of Well :

We have -

• Radius of well = 7 m.
• Depth of well = 8 m.

We know -

$\odot\;\boxed{\sf Volume_{\;Cylinder}=\pi r^2h}$

So,

$\colon \rarr \sf \: volume _{ \: well} = \frac{22}{7} \times {7}^{2} \times 8 \: {m}^{3} \\ \\ \colon \rarr \sf \: volume _{ \: well} = 22 \times 7 \times 8 \: {m}^{3} \\ \\ \colon \dashrightarrow \: \boxed{ \pink{ \frak{volume _{ \: well} = \rm 1232 \: \frak{m}^{3}}}}$

◆ Finding the Volume of the Embankment :

We know -

$\odot\;\boxed{\sf volume_{\:embankment}=\pi(R^2-r^2) h}$

Here,

• r = Radius of well = 7 m.
• R = radius of well + width of embankment = 7+21 = 28 m.
• h = height of embankment.

So,

$\colon \rarr \sf \: volume _{ \: embankment} = \frac{22}{7} \times ( {28}^{2} - {7}^{2} ) \times h \\ \\ \colon \rarr \sf \: volume _{ \: embankment} = \frac{22}{7} (784 - 49)h \\ \\ \colon \rarr \sf \: volume _{ \: embankment} = \frac{22}{ \cancel7} \times \cancel{735} \: h \\ \\ \colon \sf \rarr \: volume _{ \: embankment} = 22 \times 105 \: h \\ \\ \colon \dashrightarrow \: \boxed{ \frak{ \pink{volome _{ \: embankment} = \rm 2310 \frak{h}}}}$

◆ Finding the Height of the Embankment :

We have -

• Volume of well = 1232 m³
• Volume of embankment = 2310h m³

We know

$\odot \: \boxed{ \sf volume \: of \: well = volume \: of \: embankment}$

so,

$\colon \implies \sf \: 2310h = 1232 \\ \\ \colon \implies \sf \: h = \cancel\frac{1232}{ 2310} \\ \\ \colon \dashrightarrow \: \underline{ \boxed{ \frak{ \red{h = \frac{8}{15} \: m}}}}$

Also,

$\colon \leadsto \sf \: \frac{8}{15} \: m = \bf 0.5333 \: m \\ \\ \sf \: or \\ \\ \colon \leadsto \bf \: 53.33 \: cm$

$\therefore\;\underline{\sf The \:height\:of\:the\:embankment\:is\:\bf{53.33\:m.}}$

_____________________________

Answer 2

Answer :

Height = 0.5333 m .

or

Height = 53.33 cm.

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