NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 12: Surface Areas & Volumes - Exercise 12.2

Question 6

Step 1: The vessel has two parts:

• A cylinder of radius \(r\) and height \(h\).
• At the bottom, a hemispherical portion (half of a sphere) is raised inside the cylinder.

Step 2: Formula for volume of a cylinder:

\(V_{cylinder} = \pi r^2 h\) (in cubic metres if \(r,h\) are in metres).

Step 3: Formula for volume of a sphere:

\(V_{sphere} = \dfrac{4}{3} \pi r^3\).

Step 4: Since the bottom is a hemisphere (half of a sphere), its volume is:

\(V_{hemisphere} = \dfrac{1}{2} \times V_{sphere} = \dfrac{1}{2} \times \dfrac{4}{3}\pi r^3 = \dfrac{2}{3}\pi r^3\).

Step 5: Capacity of vessel = Volume of cylinder − Volume of hemisphere (because the hemisphere takes up space inside the cylinder).

So, \(V = \pi r^2 h - \dfrac{2}{3} \pi r^3\).

Step 6: Take \(\pi r^2\) common:

\(V = \pi r^2 \left(h - \dfrac{2}{3}r\right)\).

Step 7: Write in fraction form:

\(V = \dfrac{\pi r^2}{3} (3h - 2r)\).

Step 8: This matches exactly the given formula.

Hence, the statement is True.