Question

Earth is dug out in a cylindrical shape from a plot of land. The hole is $ 30 $ m deep and $ 7 $ m in diameter. This earth is spread out on an adjacent plot of land, which is in the shape of a rectangle of length $ 33 $ m and breadth $ 14 $ m. To what extent will the height of the rectangular plot rise?

Answer 1

Hint: In order to answer this question, we have to first find the volume of earth that is dug out. We can do this by calculating the volume of the cylinder that has been dug out. Now, this volume will be equal to the volume of the cuboid formed from building the rectangle. Calculate the height accordingly.

Complete step-by-step answer:
Given to us are the length of the cylinder $ l = 30 $ m and diameter of the cylinder $ d = 7 $ m.
We know that the radius of a cylinder is half of its diameter so the radius would be
 $ r = \dfrac{d}{2} = \dfrac{7}{2} $ m.
Now, we can write the formula for the volume of the cylinder as
$ V = \pi {r^2}h $
By substituting the above values in this formula, we get
 $ V = \dfrac{{22}}{7} \times {\left( {\dfrac{7}{2}} \right)^2} \times 30 = 1155\;{m^3} $
Now this volume of earth is being shaped into a rectangle so the volume of the rectangle should be equal to the volume of the cylinder dug out. The length and breadth of the rectangle are already given.
So the volume of the rectangle would be $ V = lbh $
By substituting the above values, we get
 $ 1155 = 33 \times 14 \times h $
On solving, we get
 $ h = \dfrac{{1155}}{{33 \times 14}} = 2.5 $ m.
Hence the height of the rectangle would be $ 2.5 $ m.
So, the correct answer is “ $ 2.5 $ m”.

Note: It should be noted that a rectangle is a two dimensional shape and the three dimensional shape of a rectangle is a cuboid. Hence in order to find the height of the rectangle we use the formula for the volume of a cuboid which is $ V = lbh $