Volume = πr²h where h = speed × time. Real-world application problem.

Water Flows Through Cylindrical Pipe — Volume per Minute

Question

Water flows through a cylindrical pipe of internal diameter 14 cm at a speed of 20 m/min. Find the volume of water (in litres) delivered by the pipe in 10 minutes.

(CBSE 2024 Board Exam)

Solution — Step by Step

Internal diameter = 14 cm, so internal radius $r = 7$ cm. We’re finding the volume of water that passes through the pipe — this is just the volume of a cylinder where the length equals the distance water travels.

Water moves at 20 m/min = 2000 cm/min. In 10 minutes:

h = 2000 \times 10 = 20{,}000 \text{ cm}

Think of it this way: if you could “freeze” all the water flowing in 10 minutes and lay it out, you’d get a cylinder 20,000 cm long with radius 7 cm.

V = \pi r^2 h = \frac{22}{7} \times 7^2 \times 20{,}000

= \frac{22}{7} \times 49 \times 20{,}000 = 22 \times 7 \times 20{,}000

= 3{,}080{,}000 \text{ cm}^3

We know that $1000 \text{ cm}^3 = 1$ litre. So:

V = \frac{3{,}080{,}000}{1000} = \boxed{3080 \text{ litres}}

Why This Works

The key insight is treating the stream of water as a cylinder. The radius of that cylinder is the pipe’s internal radius. The length (height) of that cylinder is however far the water travels in the given time.

This is why the formula is $V = \pi r^2 \times (\text{speed} \times \text{time})$ . Speed × time gives you distance — the “length” of the imaginary water cylinder. This is the standard approach for all flowing-fluid problems in Class 10.

Units are the most common source of errors here. Speed was given in metres, but radius is in centimetres. Always convert to the same unit before multiplying. Since the answer asks for litres, converting at the end (dividing by 1000) is cleaner than mid-calculation.

Alternative Method

You can find the volume per minute first, then scale up.

Volume in 1 min:

V_1 = \pi r^2 \times 2000 = \frac{22}{7} \times 49 \times 2000 = 308{,}000 \text{ cm}^3

Volume in 10 min:

V = 308{,}000 \times 10 = 3{,}080{,}000 \text{ cm}^3 = 3080 \text{ litres}

This approach is useful when the question asks for the rate in litres per minute first, then total volume. Both methods give identical answers — pick the one that feels cleaner for the numbers given.

Common Mistake

Using diameter instead of radius in the formula. Students write $V = \pi \times 14^2 \times h$ and get an answer four times too large. The formula needs $r^2$ , not $d^2$ . Always halve the diameter before squaring — and do it as the very first step so you don’t forget mid-calculation.

Unit conversion is the #1 reason for wrong answers in this question type. Write down all units explicitly as you multiply: $\frac{22}{7} \times 7^2 \text{ cm}^2 \times 20{,}000 \text{ cm}$ . When the units read cm³ at the end, you know you’re safe to convert to litres.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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