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

Given, Depth of the well$=14m$, the radius$=2m.$

The volume of earth taken out $=\pi r^{2}h$

Let $h$be the height of a well and $r$ be the radius of the well

$h=14m,r=\frac{d}{2}=2m$

Step 1: Find the volume of the given cylinder.

The volume of earth taken out after digging the well$=\pi \times r^2\times h$

$=\frac{22}{7}\times 2\times 2\times 14=176m^3$

Let $x$ be the width of the embankment formed.

Total width of well including embankment $\left(R\right)=2+x$

Height of embankment$\left(H\right)=40cm=\frac{20}{100}m\left[1cm=\frac{1}{100}m\right]$

Step 2: find the width of the embankment.

The volume of well $=$ Volume of the embankment

so, the volume of the embankment$=\pi \times (R^2-r^2)\times H=176$

$$\begin{gathered} \therefore \frac{22}{7}\times [{(2+x)}^2-{\left(2\right)}^2]\times \frac{40}{100}=176 \\ \frac{44}{35}[x^2+4x]=176 \\ 44x^2+176x=6160\left[Dividingequationby44.\right] \\ {x}^2+4x-140=0 \\ {x}^2+14x–10x-140=0\left[Splittingmiddleterm\right] \\ x(x+14)-10(x+14)=0 \\ (x+14)(x-10)=0 \\ x+14=0orx-10=0 \\ x=-14orx=10 \end{gathered}$$

Therefore, the width of the embankment is $10m$