Question

Water flows out through a circular pipe, whose internal diameter is \[1\dfrac{1}{3}\] cm, at the rate of 0.63 m per second into a cylindrical tank, the radius of whose base is 0.2 m. By how much will the level of water rise in one hour?
(a). 2.32 m
(b). 2.52 m
(c). 2.72 m
(d). 2.92 m

Answer 1

Hint: Find the volume of water that flows through the pipe in one hour and equate it to the formula of volume of the cylindrical tank, that is, \[V = \pi {r^2}h\] where ‘r’ is the radius of the base and ‘h’ is the height of the cylindrical tank.

Complete step-by-step answer:

It is given that the water flows out through a circular pipe, whose internal diameter is \[1\dfrac{1}{3}\] cm. Converting mixed fraction \[1\dfrac{1}{3}\] into an improper fraction, we have:
\[1\dfrac{1}{3} = \dfrac{{1 \times 3 + 1}}{3}\]
\[1\dfrac{1}{3} = \dfrac{4}{3}\]
The radius of the circular pipe is half of the diameter, that is, half of \[\dfrac{4}{3}\] cm.
\[{r_{pipe}} = \dfrac{1}{2} \times \dfrac{4}{3}\]
\[{r_{pipe}} = \dfrac{2}{3}cm\]
The area of cross-section of the pipe is given as \[\pi {r_{pipe}}^2\], hence, we have:
\[A = \pi {r_{pipe}}^2\]
\[A = \pi {\left( {\dfrac{2}{3}} \right)^2}\]
\[A = \dfrac{{4\pi }}{9}c{m^2}...........(1)\]
The water flows at a rate of 0.63 m per second into the cylindrical tank.
\[0.63m/s = 63cm/s\]
The volume of water flowing into the cylinder per second is equal to the product of the area of cross-section and the flow rate.
\[v = \dfrac{{4\pi }}{9} \times 63\]
\[v = \dfrac{{252\pi }}{9}c{m^3}/s\]
The volume of water that flows into the tank in one hour is given by:
\[V = \dfrac{{252\pi }}{9} \times 60 \times 60\]
\[V = 100800\pi c{m^3}\]
Converting volume into cubic meters by dividing by \[{10^6}\].
\[V = \dfrac{{100800\pi }}{{{{10}^6}}}{m^3}\]
\[V = 0.1008\pi {m^3}...........(2)\]
We know that the volume of a cylinder with radius r and height h is given as follows:
\[V = \pi {r^2}h\]
The radius of the base of the cylindrical tank is 0.2 m.
\[V = \pi {(0.2)^2}h\]
\[V = 0.04\pi h...........(3)\]
We know that equation (2) and equation (3) are equal. Then, we have:
\[0.04\pi h = 0.1008\pi \]
Solving for h, we have:
\[h = \dfrac{{0.1008}}{{0.04}}\]
\[h = 2.52m\]
Hence, the water fills up to a height of 2.52 m.
Hence, the correct answer is option (b).

Note: Convert all quantities into the same units before multiplying or dividing them, otherwise, the answer will differ by powers of 10, which is wrong.