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Surface Areas and Volumes
RD Sharma
CBSE
Grade 10
Question:
A wooden toy rocket is in the shape of a cone mounted on a cylinder. The height of the entire rocket is $26\text{ cm}$, while the height of the conical part is $6\text{ cm}$. The base of the conical portion has a diameter of $5\text{ cm}$, while the base diameter of the cylindrical portion is $3\text{ cm}$. If the conical portion is to be painted orange and the cylindrical portion yellow, find the area of the rocket painted with each of these colours. (Take $\pi = 3.14$)
Step-by-Step Solution
Key Concept: Cone: $r_1 = 2.5\text{ cm}, h_1 = 6\text{ cm} \Rightarrow l = \sqrt{36 + 6.25} = 6.5\text{ cm}$. Cylinder: $r_2 = 1.5\text{ cm}, h_2 = 20\text{ cm}$.<br>Orange Area (Cone) $= \text{CSA cone} + \text{Base area cone} - \text{Base area cylinder} = \pi r_1 l + \pi r_1^2 - \pi r_2^2 = 3.14 \times 2.5 \times 6.5 + 3.14 \times (6.25 - 2.25) = 51.025 + 12.56 = 63.585\text{ cm}^2$.<br>Yellow Area (Cylinder) $= \text{CSA cylinder} + \text{Base area cylinder} = 2 \pi r_2 h_2 + \pi r_2^2 = 3.14 \times 1.5 \times (40 + 1.5) = 4.71 \times 41.5 = 195.465\text{ cm}^2$.
Cone slant height $l = \sqrt{6^2 + 2.5^2} = 6.5\text{ cm}$. [1.0 Mark]
Orange Area $= \text{CSA cone} + \pi(r_1^2 - r_2^2) = 3.14 \times 2.5 \times 6.5 + 3.14(6.25 - 2.25) = 51.025 + 12.56 = 63.585\text{ cm}^2$. [2.0 Marks]
Yellow Area $= 2 \pi r_2 h_2 + \pi r_2^2 = 3.14 \times 1.5 \times 40 + 3.14 \times 2.25 = 188.4 + 7.065 = 195.465\text{ cm}^2$. [2.0 Marks]
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🎯 Official CBSE Marking Scheme:
Calculating cone slant height $l = 6.5\text{ cm}$: 1.0 Mark
Evaluating Orange painted area $= 63.585\text{ cm}^2$: 2.0 Marks
Evaluating Yellow painted area $= 195.465\text{ cm}^2$: 2.0 Marks
Orange Area $= \text{CSA cone} + \pi(r_1^2 - r_2^2) = 3.14 \times 2.5 \times 6.5 + 3.14(6.25 - 2.25) = 51.025 + 12.56 = 63.585\text{ cm}^2$. [2.0 Marks]
Yellow Area $= 2 \pi r_2 h_2 + \pi r_2^2 = 3.14 \times 1.5 \times 40 + 3.14 \times 2.25 = 188.4 + 7.065 = 195.465\text{ cm}^2$. [2.0 Marks]
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🎯 Official CBSE Marking Scheme:
Calculating cone slant height $l = 6.5\text{ cm}$: 1.0 Mark
Evaluating Orange painted area $= 63.585\text{ cm}^2$: 2.0 Marks
Evaluating Yellow painted area $= 195.465\text{ cm}^2$: 2.0 Marks
Correct Answer:
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