Question 343 marksShort AnswerSection C

A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use π=227\pi = \frac{22}{7})

Surface Areas and Volumes
Volume of a Combination of Solids

Official answer

Total volume ≈ 680.17 cm3680.17\text{ cm}^3 (exact value 40816 cm3\frac{4081}{6}\text{ cm}^3). With radius r=3.5 cmr = 3.5\text{ cm}, cylinder height h=20−7=13 cmh = 20-7 = 13\text{ cm}; Volume=πr2h+43πr3=500.5+179.67\text{Volume} = \pi r^2 h + \frac{4}{3}\pi r^3 = 500.5 + 179.67 ≈ 680.17 cm3680.17\text{ cm}^3.

Answer keywords:cylinder with hemispherical endscombination of solidsvolume of cylindervolume of hemispherecapsule shapemensuration

Marking scheme

  1. 1.1 mark: correctly finding radius r=3.5 cmr = 3.5\text{ cm} and cylinder height h=20−2(3.5)=13 cmh = 20 - 2(3.5) = 13\text{ cm}.
  2. 2.1 mark: correct formula setup — Total Volume=πr2h+2×23πr3\text{Total Volume} = \pi r^2 h + 2\times\frac{2}{3}\pi r^3 (cylinder + two hemispherical ends).
  3. 3.1 mark: correct substitution and final numeric answer ≈ 680.17 cm3680.17\text{ cm}^3 (or equivalent exact fraction 40816 cm3\frac{4081}{6}\text{ cm}^3), with proper units.

Hint

Radius=diameter2=3.5 cm\text{Radius} = \frac{\text{diameter}}{2} = 3.5\text{ cm}; height of cylinder = total height − 2×radius; then Volume=πr2h (cylinder)+2×23πr3 (two hemispheres)\text{Volume} = \pi r^2 h \text{ (cylinder)} + 2\times\frac{2}{3}\pi r^3 \text{ (two hemispheres)}.

Quick oral answer

Find radius 3.5 cm from diameter 7 cm, subtract 2×radius from total height 20 cm to get cylinder height 13 cm, then add the cylinder's volume πr2h\pi r^2 h to the volume of two hemispheres 23πr3\frac{2}{3}\pi r^3 each, giving about 680.17 cm³.

Common mistakes

  1. 1.Subtracting 2× the diameter instead of 2× the radius from the total height to find the cylinder's height, leading to a negative or wrong cylinder height.
  2. 2.Using the full sphere volume formula 43πr3\frac{4}{3}\pi r^3 instead of the hemisphere formula 23πr3\frac{2}{3}\pi r^3 for each rounded end.
  3. 3.Arithmetic slips when multiplying fractions with π=227\pi=\frac{22}{7} across multiple terms, especially forgetting to multiply by 2 for the two hemispherical ends.

Analysis and explanation

A combination-solids mensuration problem: a cylinder capped by two hemispheres, forming a capsule shape.


Concept

  • Each hemisphere adds height equal to its radius (not diameter), so Total height=cylinder height+2r\text{Total height} = \text{cylinder height} + 2r.
  • Total Volume=Volume of cylinder+2×Volume of hemisphere=πr2h+43πr3\text{Total Volume} = \text{Volume of cylinder} + 2\times\text{Volume of hemisphere} = \pi r^2 h + \frac{4}{3}\pi r^3.

Key points

  • Radius r=diameter2=3.5 cmr = \frac{\text{diameter}}{2} = 3.5\text{ cm}; cylinder height h=20−2(3.5)=13 cmh = 20 - 2(3.5) = 13\text{ cm}.
  • Use π=227\pi = \frac{22}{7} throughout, keeping fractions exact until the final step to avoid rounding drift.

Common mistakes

  • Subtracting 2×diameter instead of 2×radius from the total height.
  • Rounding intermediate hemisphere volumes too early, causing accumulated error.

Real-world

  • This capsule shape models medicine capsules, storage tanks and test tubes with domed ends.

Interesting facts

This 'cylinder with hemispherical ends' shape is called a capsule or 'spherocylinder' in geometry, and is exactly the shape of medicine capsules, biological cells like certain bacteria (e.g., E. coli), and pressure vessels in engineering.

Designing storage tanks and pressure vessels with hemispherical (domed) ends rather than flat ends is a real engineering practice because domed ends distribute internal pressure more evenly, reducing stress concentration.

The volume formula for this composite solid — πr2(h+4r3)\pi r^2\left(h + \frac{4r}{3}\right) — reduces to the volume of a pure sphere 43πr3\frac{4}{3}\pi r^3 if the cylindrical height h is set to zero, showing the sphere is a limiting case of this shape.

Question details

Q34 of 56 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Standard)
Year
2026
Section
Section C
Marks
3
Type
Short Answer
Chapter
Surface Areas and Volumes
Topic
Volume of a Combination of Solids
Bloom's level
apply
Syllabus topic
Unit 7: Mensuration — Surface Areas and Volumes: Combination of a Cylinder and Two Hemispheres
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

Why do we subtract 2×radius (not 2×diameter) from the total height?

Because each hemispherical end adds a height equal to its radius (a hemisphere's 'height' from its flat circular base to its curved top equals the radius), not its diameter. So Total height=cylinder height+radius (top)+radius (bottom)=cylinder height+2r\text{Total height} = \text{cylinder height} + \text{radius (top)} + \text{radius (bottom)} = \text{cylinder height} + 2r.

What is the general formula for the volume of this composite solid?

Total Volume=πr2h+43πr3=πr2(h+4r3)\text{Total Volume} = \pi r^2 h + \frac{4}{3}\pi r^3 = \pi r^2\left(h + \frac{4r}{3}\right), where r is the common radius of the cylinder and hemispheres, and h is the height of only the cylindrical part.