A solid consists of a cylinder with a hemisphere at one end and a cone at the other end. Their common radius is $7\text{ cm}$. The height of the cylinder is $20\text{ cm}$ and height of the cone is $24\text{ cm}$. Find the total surface area and total volume of the solid.
Step-by-Step Solution
Key Concept: $r = 7\text{ cm}, h_{cyl} = 20\text{ cm}, h_{cone} = 24\text{ cm} \Rightarrow l_{cone} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = 25\text{ cm}$.<br>TSA $= \text{CSA cone} + \text{CSA cylinder} + \text{CSA hemisphere} = \pi r l + 2 \pi r h_{cyl} + 2 \pi r^2 = \pi r(l + 2h_{cyl} + 2r) = \dfrac{22}{7} \times 7 \times (25 + 40 + 14) = 22 \times 79 = 1738\text{ cm}^2$.<br>Total Vol $= \dfrac{1}{3} \pi r^2 h_{cone} + \pi r^2 h_{cyl} + \dfrac{2}{3} \pi r^3 = \pi r^2 \left(\dfrac{h_{cone}}{3} + h_{cyl} + \dfrac{2r}{3}\right) = \dfrac{22}{7} \times 49 \times \left(8 + 20 + \dfrac{14}{3}\right) = 154 \times \dfrac{98}{3} = \dfrac{15092}{3} = 5030.67\text{ cm}^3$.
Cone slant height $l = \sqrt{24^2 + 7^2} = 25\text{ cm}$. [1.0 Mark]
$\text{TSA} = \pi r(l + 2h + 2r) = 22 \times (25 + 40 + 14) = 22 \times 79 = 1738\text{ cm}^2$. [2.0 Marks]
$\text{Total Vol} = 154 \times \left(8 + 20 + \dfrac{14}{3}\right) = 154 \times \dfrac{98}{3} = 5030.67\text{ cm}^3$. [2.0 Marks]
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🎯 Official CBSE Marking Scheme:
Calculating slant height $l = 25\text{ cm}$: 1.0 Mark
Evaluating total surface area $= 1738\text{ cm}^2$: 2.0 Marks
Evaluating total volume $= 5030.67\text{ cm}^3$: 2.0 Marks
Correct Answer: