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Surface Areas and Volumes
NCERT Exemplar
CBSE
Grade 10

Question:

Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminum sheet. The diameter of the model is $3\text{ cm}$ and its length is $12\text{ cm}$. If each cone has a height of $2\text{ cm}$, find the volume of air contained in the model that Rachel made.

Step-by-Step Solution

Key Concept: Radius $r = 1.5\text{ cm}$. Conical height $h_1 = 2\text{ cm}$. Cylindrical height $h_2 = 12 - 2(2) = 8\text{ cm}$. Total volume $= \text{Vol(cylinder)} + 2 \times \text{Vol(cone)}$.
Radius $r = 1.5 = 3/2\text{ cm}$. Height of cylinder $h_2 = 12 - 4 = 8\text{ cm}$. Height of cone $h_1 = 2\text{ cm}$. [1.0 Mark]
Volume of cylinder $= \pi r^2 h_2 = \dfrac{22}{7} \times \left(\dfrac{3}{2}\right)^2 \times 8 = \dfrac{22}{7} \times \dfrac{9}{4} \times 8 = \dfrac{396}{7}\text{ cm}^3$. [1.0 Mark]
Volume of 2 cones $= 2 \times \left[\dfrac{1}{3} \pi r^2 h_1\right] = \dfrac{2}{3} \times \dfrac{22}{7} \times \dfrac{9}{4} \times 2 = \dfrac{66}{7}\text{ cm}^3$.
Total volume $= \dfrac{396}{7} + \dfrac{66}{7} = \dfrac{462}{7} = 66\text{ cm}^3$. [1.0 Mark]

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🎯 Official CBSE Marking Scheme:
Finding cylinder height $8\text{ cm}$: 1.0 Mark
Calculating cylinder volume $= 396/7\text{ cm}^3$: 1.0 Mark
Calculating 2 cones volume $= 66/7\text{ cm}^3$ and total $= 66\text{ cm}^3$: 1.0 Mark

Correct Answer:
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