NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 12: Surface Areas & Volumes - Exercise 12.4

Question 5

Step 1: Write down the given data.

• Speed of water = \(10\,\text{m/min} = 1000\,\text{cm/min}\) (since \(1\,\text{m} = 100\,\text{cm}\)).
• Pipe diameter = \(5\,\text{mm} = 0.5\,\text{cm}\). Radius of pipe = \(0.25\,\text{cm}\).
• Conical vessel diameter = \(40\,\text{cm}\). Radius of cone = \(20\,\text{cm}\).
• Height (depth) of cone = \(24\,\text{cm}\).

Step 2: Find the volume of water flowing through the pipe per minute (flow rate).

The volume of water coming out in one minute = cross-sectional area of pipe × speed of flow.

Cross-sectional area of pipe = \(\pi r^2 = \pi (0.25)^2 = 0.0625\pi\,\text{cm}^2\).

So, volume per minute = \(0.0625\pi \times 1000 = 62.5\pi\,\text{cm}^3/ ext{min}\).

Step 3: Find the volume of the conical vessel.

Formula: \(V = \tfrac{1}{3}\pi r^2 h\).

Here, \(r = 20\,\text{cm}, h = 24\,\text{cm}\).

So, \(V = \tfrac{1}{3} \pi (20)^2 (24) = \tfrac{1}{3} \pi (400)(24) = 3200\pi\,\text{cm}^3\).

Step 4: Find the time required.

Time = Total volume ÷ Volume per minute

\(t = \dfrac{3200\pi}{62.5\pi} = \dfrac{3200}{62.5} = 51.2\,\text{minutes}\).

Final Answer: The conical vessel will be filled in 51.2 minutes.