Answer
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Hint: Find the volume of water that flows through the pipe in half an 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 2 cm.
The radius of the circular pipe is half of the diameter, that is, half of 2 cm.
\[{r_{pipe}} = \dfrac{1}{2} \times 2\]
\[{r_{pipe}} = 1cm\]
The area of cross-section of the pipe is given as \[\pi {r_{pipe}}^2\], hence, we have:
\[A = \pi {r_{pipe}}^2\]
\[A = \pi {1^2}\]
\[A = \pi c{m^2}...........(1)\]
The water flows at a rate of 7 m per second into the cylindrical tank.
\[7m/s = 700cm/s\]
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.
\[v = \pi \times 700\]
\[v = 700\pi c{m^3}/s\]
The volume of water that flows into the tank in half an hour is given by:
\[V = 700\pi \times \dfrac{1}{2} \times 60 \times 60\]
\[V = 1260000\pi c{m^3}............(2)\]
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 40 cm.
\[V = \pi {(40)^2}h\]
\[V = 1600\pi h...........(3)\]
We know that equation (2) and equation (3) are equal. Then, we have:
\[1600\pi h = 1260000\pi \]
Solving for h, we have:
\[h = \dfrac{{1260000}}{{1600}}\]
\[h = 787.5m\]
Hence, the water fills up to a height of 787.5 cm.
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.
Complete step by step answer:
It is given that the water flows out through a circular pipe, whose internal diameter is 2 cm.
The radius of the circular pipe is half of the diameter, that is, half of 2 cm.
\[{r_{pipe}} = \dfrac{1}{2} \times 2\]
\[{r_{pipe}} = 1cm\]
The area of cross-section of the pipe is given as \[\pi {r_{pipe}}^2\], hence, we have:
\[A = \pi {r_{pipe}}^2\]
\[A = \pi {1^2}\]
\[A = \pi c{m^2}...........(1)\]
The water flows at a rate of 7 m per second into the cylindrical tank.
\[7m/s = 700cm/s\]
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.
\[v = \pi \times 700\]
\[v = 700\pi c{m^3}/s\]
The volume of water that flows into the tank in half an hour is given by:
\[V = 700\pi \times \dfrac{1}{2} \times 60 \times 60\]
\[V = 1260000\pi c{m^3}............(2)\]
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 40 cm.
\[V = \pi {(40)^2}h\]
\[V = 1600\pi h...........(3)\]
We know that equation (2) and equation (3) are equal. Then, we have:
\[1600\pi h = 1260000\pi \]
Solving for h, we have:
\[h = \dfrac{{1260000}}{{1600}}\]
\[h = 787.5m\]
Hence, the water fills up to a height of 787.5 cm.
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.
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