Question

Selvi’s house has an overhead tank in the shape of a cylinder. This is filled by pumping water from a sump (underground tank), which is in the shape of a cuboid. The sump has a dimension 1.57m x 1.44m x 0.95m. The overhead tank has its radius of 60 cm and its height is 95 cm. Find the height of the water, left in the sump after the overhead tank has been completely filled with water from a sump which had been full. Compare the capacity of the tank with that of the sump (use $\pi =\dfrac{22}{7}$ ).

Answer 1

Hint: Capacity of the overhead tank is the volume of water it can store. So, we have to determine the volume of the overhead tank, i.e. a cylinder is given by $V=\pi {{r}^{2}}h$ and also the water capacity of the sump, i.e. a cuboid given by ${{V}_{c}}=l\times b\times h$.

We know that Selvi’s house has an overhead tank in the shape of a cylinder. This is filled by pumping water from a sump (underground tank), which is in the shape of a cuboid.
Now, we know that the overhead tank is in the shape of a cylinder whose radius is given as r = 60 cm and its height is given as h = 95 cm.
We know that the water holding capacity of the overhead tank is equal to the volume of the cylinder.
We also know that the volume of the cylinder is equal to $\pi {{r}^{2}}h$. So, let the volume of the cylindrical tank be V. Then we can write,
$V=\pi {{r}^{2}}h$
Putting the value of r and h, we get,
$V=\pi {{\left( 60 \right)}^{2}}\left( 95 \right)$
Using the value of $\pi =\dfrac{22}{7}$ as mentioned in the question, we get
$V=\dfrac{22}{7}\left( 3600 \right)\left( 95 \right)$
It is given in cubic cm, so we have to convert it into the cubic meter as below,
$V=\dfrac{22}{7}\left( 3600 \right)\left( 95 \right)\dfrac{1}{1000000}$
$V=\dfrac{22}{7}\left( 0.36 \right)\left( 0.95 \right)\text{ meter cubic}....\left( i \right)$
Now we know that the cylindrical overhead tank is filled by the sump, which is in cuboidal shape. The dimensions of the sump i.e. in the shape of the cuboid are given as 1.57m x 1.44m x 0.95m. Therefore, we can write that
Length of the cuboidal tank, l = 1.57 m.
The breadth of the cuboidal tank, b = 1.44 m.
Height of the cuboidal tank, h = 0.95 m.
The capacity of the tank is equal to the volume of the cuboid.
Let the volume of the cuboid be ${{V}_{c}}$. Then, we can write it as,
${{V}_{c}}=l\times b\times h$
Substituting the values, we get
${{V}_{c}}=1.57\times 1.44\times 0.95....\left( ii \right)$
Let the height of the water left in the sump after filling the overhead tank be ${{h}_{o}}$. Breadth and length of cuboid would remain the same after defiling the tank.
So, we can write that the volume of water left in the sump
\[=l\times b\times {{h}_{o}}=1.57\times 1.44\times {{h}_{o}}....\left( iii \right)\]
The volume of the water consumed from the cuboidal tank (sump) = volume of the cylindrical tank = V
So, we get,
Water left in the sump = volume of the cuboid - the volume of water consumed
Water left in the sump $={{V}_{c}}-V$
From (i) , (ii) and (iii)
$1.57\times 1.44\times {{h}_{o}}=1.57\times 1.44\times 0.95-\left( \dfrac{22}{7}\left( 0.36 \right)\left( 0.95 \right) \right)$
Now solving the equation to get the value of ${{h}_{o}}$, we get
$1.57\times 1.44\times {{h}_{o}}=1.0729$
${{h}_{o}}=0.47456m$
Rounding off up to two terms of decimal, we can write the height as
${{h}_{o}}=0.47m$
The height of the water left in the sump after the overhead tank has been filled with water from a sump that had been full is 0.47m or 47 cm.
Now we need to compare the capacity of the tank with that of the sump.
We know that,
The volume of the water consumed from the cuboidal tank (sump) = volume of the cylindrical tank
The volume of water consumed from the sump $=l\times b\times \left( 0.95-{{h}_{o}} \right)$
We have found the value of ${{h}_{o}}$, so we can substitute it in the above equation as,
The volume of water consumed from the sump $=l\times b\times \left( 0.95-0.47 \right)$
On comparing it with the capacity of the sump (volume of the cuboidal tank) $=\dfrac{l\times b\times (0.95-0.47)}{l\times b\times 0.95}$, l, b will get canceled and we will get the ratio as,
$=\dfrac{0.48}{0.95}=0.5052$
So, we can say that the capacity of the overhead tank is 0.5 of the capacity of the sump.
Therefore, the capacity of the overhead tank is half of the capacity of the sump.

Note: We can compare the water of the overhead tank with the sump in an alternate way also i.e. by taking directly into consideration the volume of the overhead tank which is in the form of a cylinder, so now we will compare as \[\dfrac{\dfrac{22}{7}(0.36)(0.95)}{1.57\times 1.44\times 0.95}\] canceling 0.95 and evaluating, we will get it as 0.5. Therefore, the capacity of the overhead tank is half of the capacity of the sump.