Question

Find the rise of water level in the tank due to rainwater, if rain falls\[100cm\] , and it falls on a roof of a \[50m\] long and \[30m\] wide, and collected in a cylinder tank of \[21m\] the radius.
\[
  {\text{A}}{\text{. }}1.08m \\
  {\text{B}}{\text{. }}1m \\
  {\text{C}}{\text{. }}1.4m \\
  {\text{D}}{\text{. }}1.88 \\
 \]

Answer 1

Hint: Volume of the waterfall can be calculated for the roof as the height of rainfall is given as\[100cm\] rainfall that is the depth of the rainfall on a plane surface of the roof. The total volume of rainfall on a roof is transferred to a cylinder tank where a rise in the level of rainfall is calculated.

Complete step by step answer:
Since the length and width of the roof is given the volume of rainfall can be calculated as height of rainfall is also given, the length, width, and height of rainfall on the roof makes it to act like a cuboid hence, the volume of the cuboid is given as\[V = L \times W \times H\], where
Length\[L = 50m\],
Width\[W = 30m\],
Height\[H = \dfrac{{100}}{{100}}cm = 1m\]
Now, the volume of rainfall is:
\[
  V = L \times W \times H \\
   = 50 \times 30 \times 1 \\
   = 1500{m^3} \\
 \]
The volume of the waterfall \[1500{m^3}\]will now be the volume of water transferred to the tank. This water transferred to the tank will raise the level of water in the tank.
The volume of the waterfall on the roof = volume of water in the tank.
\[1500{m^3} = \pi {r^2}h\]
Where, \[r\]is the radius of tank given as,\[r = 21m\]
\[h\], being the height of raise in water to be calculated, by cross multiplying the terms of the equation \[1500{m^3} = \pi {r^2}h\], we get the height raised as:
   \[
  h = \dfrac{{1500}}{{\pi {r^2}}} \\
   = \dfrac{{1500}}{{\pi \times {{21}^2}}} \\
   = \dfrac{{1500}}{{\dfrac{{22}}{7} \times 21 \times 21}} \\
   = \dfrac{{1500}}{{1386}} \\
   = 1.08m \\
 \]
Hence, the rise in the water level of the water tank is\[1.08m\].

Note: In general, the amount of rainfall is expressed in centimeters (millimeters), which express the depth or the height of the waterfall on a level surface, which is used to determine the quantity of the rainfall.