A rocket is a cylinder (radius 3 cm, height 12 cm) surmounted by a cone of the same radius and slant height 5 cm. Find the total surface area and volume. [Use \(\pi=3.14\)].
TSA = \(96\pi\approx 301.44\,\text{cm}^2\); Volume = \(120\pi\approx 376.8\,\text{cm}^3\)
Step 1: Write down the given dimensions
Step 2: Find the vertical height of the cone
The cone’s vertical height \(h_{\rm cone}\) can be found using Pythagoras theorem:
\[ h_{\rm cone} = \sqrt{l^2 - r^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4\,\text{cm}. \]
Step 3: Total Surface Area (TSA)
TSA includes:
So,
\[ TSA = 2\pi r h_{\rm cyl} + \pi r^2 + \pi r l \]
Substitute values:
\[ TSA = 2 \times 3.14 \times 3 \times 12 + 3.14 \times 3^2 + 3.14 \times 3 \times 5 \]
\[ = 226.08 + 28.26 + 47.1 = 301.44\,\text{cm}^2 \]
Step 4: Volume
Total volume = Volume of cylinder + Volume of cone
\[ V = \pi r^2 h_{\rm cyl} + \tfrac{1}{3}\pi r^2 h_{\rm cone} \]
Substitute values:
\[ V = 3.14 \times 3^2 \times 12 + \tfrac{1}{3} \times 3.14 \times 3^2 \times 4 \]
\[ = 339.12 + 37.68 = 376.8\,\text{cm}^3 \]
Final Answer:
Total Surface Area = 301.44 cm²
Volume = 376.8 cm³