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Areas Related To Circles
EXAMPLES
CBSE_NCERT_TEXTBOOK
Grade 10
Question:
A wooden toy rocket is in the shape of a cone mounted on a cylinder, as shown in Fig. 12.8. The height of the entire rocket is 26 cm, while the height of the conical part is 6 cm. The base of the conical portion has a diameter of 5 cm, while the base diameter of the cylindrical portion is 3 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 = 3.14)
Step-by-Step Solution
Key Concept: Use the curved surface area formulas: \(\text{CSA of cone}=\pi r l\) where \(l=\sqrt{r^{2}+h^{2}}\), and \(\text{CSA of cylinder}=2\pi r h\). For the cylinder also add the area of its lower base (\(\pi r^{2}\)) because the upper base is attached to the cone and is not painted.
1. Given data - Height of whole rocket = 26 cm - Height of cone (h_c) = 6 cm - Height of cylinder (h_{cy}) = 26 – 6 = 20 cm - Diameter of cone = 5 cm \(\Rightarrow\) radius of cone \(r_c = \frac{5}{2}=2.5\) cm - Diameter of cylinder = 3 cm \(\Rightarrow\) radius of cylinder \(r_{cy}=\frac{3}{2}=1.5\) cm
2. Slant height of the cone \[ l = \sqrt{r_c^{2}+h_c^{2}} = \sqrt{(2.5)^{2}+6^{2}} = \sqrt{6.25+36}=\sqrt{42.25}=6.5\text{ cm}\]
3. Area to be painted orange (cone) Only the curved surface of the cone is exposed. \[ \text{CSA}_{cone}=\pi r_c l = 3.14 \times 2.5 \times 6.5 = 3.14 \times 16.25 = 51.055\text{ cm}^{2} \] Rounded to one decimal place: 51.1 cm².
4. Area to be painted yellow (cylinder) - Curved surface area of cylinder: \[ \text{CSA}_{cyl}=2\pi r_{cy} h_{cy}=2 \times 3.14 \times 1.5 \times 20 = 6.28 \times 30 = 188.4\text{ cm}^{2} \] - Lower base of cylinder (the upper base is attached to the cone and is not painted): \[ \text{Base area}=\pi r_{cy}^{2}=3.14 \times (1.5)^{2}=3.14 \times 2.25 = 7.065\text{ cm}^{2} \] - Total yellow area: \[ \text{Yellow area}=188.4 + 7.065 = 195.465\text{ cm}^{2} \] Rounded to one decimal place: 195.5 cm².