Question

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

Answer 1

Answer:

Heyaa♡♡

Depth of the well :-

$h1 = 14m$

Radius of circular end of the well :-

$r1 = \frac{4}{2} \\ = > 2m$

Height of embankment :-

$h2 = 40cm \\ = > 0.4m$

Let the width of the embankment be x.

From the fig. we can see that embankment will be cylindrical in shape having outer radius (r2) as ( 2 + x) m and inner radius (r1) as 2m.

Now, Volume of earth dug from the well = Volume of earth used to for embankment.

$= > \pi \: r {1}^{2}h1 = \pi \: (r {1}^{2} - r {2}^{2} )h2 \\ = > \pi \: (2) {}^{2} 14 = \pi ((2 + x) {}^{2} - 2 {}^{2} )0.4 \\ = > 4 \times 14 = \frac{x(x + 4)4}{10} \\ = > x {}^{2} + 4x - 140 = 0 \\ = > x {}^{2} + 14x - 10x - 140 = 0 \\ = > x(x + 14) - 10(x + 14) = 0 \\ = > (x - 10)(x + 14) = 0 \\ = > x = 10$

Because x cannot be negative.

Therefore the width of the embankment will be 10m.

Hope it helped you.

Answer 2

AnsweR :

$\bf{\Large{\underline{\sf{Given\::}}}}$

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

$\bf{\Large{\underline{\sf{To\:find\::}}}}$

The width of the embankment.

$\bf{\Large{\underline{\tt{\purple{Explanation\::}}}}}$

Let the width of the embankment be R m

$\bf{\red{We\:have}\begin{cases}\sf{Diameter\:(d)\:=\:4m}\\ \sf{Radius\:(r)\:=\:2m}\\ \sf{Height\:of\:well\:(h)\:=\:14m}\\ \sf{Height\:of\:embankment\:(h)\:=\:40cm=0.4m}\end{cases}}$

Formula Use :

$\bf{\Large{\boxed{\sf{Volume\:of\:cylinder\:=\:\pi r^{2} h}}}}}$

$\leadsto\sf{Volume\:of\:the\:earth\:taken\:out\:=\:\pi r^{2} h}\\\\\\\leadsto\sf{Volume\:of\:the\:earth\:taken\:out\:=\:\bigg(\dfrac{22}{\cancel{7}} *2*2*\cancel{14}\bigg)m^{3} }\\\\\\\leadsto\sf{Volume\:of\:the\:earth\:taken\:out\:=\:\bigg(22*2*2*2\bigg)m^{3} }\\\\\\\leadsto\sf{\orange{Volume\:of\:the\:earth\:taken\:out\:=\:176m^{3} }}$

Now,

Total width of well cover embankment = (2+R) m

$\implies\sf{Volume\:of\:the\:earth\:out\:=\:\pi (R^{2} -r^{2} )h}\\\\\\\implies\sf{176=\dfrac{22}{7} *\bigg[(2+R)^{2} -2^{2} \bigg]*(0.4)}\\\\\\\implies\sf{\cancel{176}=\dfrac{\cancel{22}}{7} *\bigg[(2+R)^{2} -4\bigg]*\dfrac{4}{10} }\\\\\\\implies\sf{8=\dfrac{4}{70} *\bigg[(2+R)^{2}-4\bigg]} \\\\\\\implies\sf{\cancel{560}=\cancel{4}*\bigg[(2+R)^{2}-4\bigg]} \\\\\\\implies\sf{140\:=\:\bigg[(2+R)^{2} -4\bigg]}\\\\\\\implies\sf{(2+R)^{2} -4-140=0}\\\\\\\implies\sf{(2+R)^{2} -144=0}$

$\implies\sf{(2+R)^{2} -12^{2} =0}\\\\\\\implies\sf{(2+R+12)(2+R-12)=0\:\:\:\:\:\:\:\:\:\:\:\big[a^{2} -b^{2}=(a+b)(a-b)\big] }\\\\\\\implies\sf{(R+14)(R-10)=0}\\\\\\\implies\sf{R+14=0\:\:\:\:\:\:\:Or\:\:\:\:\:\:\:R-10=0}\\\\\\\implies\sf{\red{R\:=\:-14\:\:\:\:\:\:\:\:\:\:Or\:\:\:\:\:\:\:\:\:R=10}}$

∴We know that negative value is not acceptable.

Thus,

The width of the embankment is 10 m.