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Areas Related To Circles
EXERCISE 12.2
CBSE_NCERT_TEXTBOOK
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 aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.) Fig. 12.14 170
Step-by-Step Solution
Key Concept: The volume of a composite solid is obtained by adding the volumes of its individual parts. Use the standard formulas: \(V_{cylinder}=\pi r^{2}h\) and \(V_{cone}=\frac{1}{3}\pi r^{2}h\).
1. Identify the dimensions - Diameter = 3 cm \(\Rightarrow\) radius \(r = \frac{3}{2}=1.5\) cm. - Height of the cylindrical part = 12 cm. - Height of each cone = 2 cm.
2. Volume of the cylindrical part \[V_{cyl}=\pi r^{2}h = \pi (1.5)^{2}(12) = \pi \times 2.25 \times 12 = 27\pi\ \text{cm}^{3}.\]
3. Volume of one cone \[V_{cone}=\frac{1}{3}\pi r^{2}h = \frac{1}{3}\pi (1.5)^{2}(2) = \frac{1}{3}\pi \times 2.25 \times 2 = 1.5\pi\ \text{cm}^{3}.\]
4. Volume of two cones \[V_{2\,cones}=2\times 1.5\pi = 3\pi\ \text{cm}^{3}.\]
5. Total volume of the model \[V_{total}=V_{cyl}+V_{2\,cones}=27\pi+3\pi=30\pi\ \text{cm}^{3}.\]
6. Numerical value (optional) Using \(\pi \approx 3.14\) or \(\frac{22}{7}\): \[V_{total}\approx 30\times 3.14 = 94.2\ \text{cm}^{3}\] (or \(\frac{660}{7}\approx 94.3\ \text{cm}^{3}\)).
Correct Answer:30π cm³ (≈ 94.2 cm³)
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