Question

A 20 meters deep well with the diameter 7 meters is dug up, and the earth from the digging is spread evenly to form a platform $22{\text{ m}} \times 14{\text{ m}}$. Determine the height of the platform.

Answer 1

Hint: In this question, we need to determine the height of the platform that is being prepared from the volume of the earth dug-out. For this, we will equate the volume of the earth dug out and the volume of the platform.

Complete step-by-step answer:
Here, as the volume of the earth that is dug up is distributed in the land so the volume of the earth (sand) will remain the same. Mathematically ${V_{cylinder}} = {V_{cuboid}}$. So, the volume of the sand dug up from the cylinder of height 20 meters and radius 7 meters will be equal to the volume of the cuboid of length 22 meters and width 14 meters.
Let the height of the cuboid be ‘h’ meters.

Now, the volume of the cylinder is the product of the square of the radius of the base and the height of the cylinder with the constant term pi. Mathematically ${V_{cylinder}} = \pi {r^2}h$.
Substitute $r = 7{\text{ m and }}h = 20{\text{ m}}$ in the formula ${V_{cylinder}} = \pi {r^2}h$ to determine the volume of the earth dugout.
$
  {V_{cylinder}} = \pi {r^2}h \\
   = \dfrac{{22}}{7} \times {(7)^2} \times 20 \\
   = \dfrac{{22}}{7} \times 7 \times 7 \times 20 \\
   = 22 \times 7 \times 20 \\
   = 3080{\text{ }}{{\text{m}}^3} \\
 $

Now, the volume of the cuboid is the product of the length, width and height of the cuboid. Mathematically, ${V_{cuboid}} = l \times b \times h$.
Substitute $l = 22{\text{ m and }}b = 14{\text{ m}}$ in the formula ${V_{cuboid}} = l \times b \times h$ to determine the volume of the dug-out earth that is uniformly spread on the earth.
$
  {V_{cuboid}} = l \times b \times h \\
   = 22 \times 14 \times h \\
   = 308h{\text{ }}{{\text{m}}^3} \\
 $

As here the volume of the cylinder and the cuboid is same so, substitute ${V_{cylinder}} = 3080{\text{ }}{{\text{m}}^3}{\text{ and }}{V_{cuboid}} = 308h{\text{ }}{{\text{m}}^3}$ in the equation ${V_{cylinder}} = {V_{cuboid}}$ to determine the height of the cuboid.
$
  {V_{cylinder}} = {V_{cuboid}} \\
  3080 = 308h \\
  h = \dfrac{{3080}}{{308}} \\
   = 10{\text{ m}} \\
 $
Hence, the height of the platform being dug out from the earth is 10 meters.

Note: Students should note here that it is not given in the question that the platform is in cuboid shape; however, it is being given that the length and the width of the platform being spread are different and so, the platform must be a cuboid.