From a solid cylinder of height $30$ cm and radius $7$ cm, a conical cavity of height $24$ cm and same radius is hollowed out. Find the total surface area of the remaining solid.
Generated by claude-sonnet-4-6 · 2026-06-15 10:34 · grounding rag
Model Answer
Given: Solid cylinder: height (H) = 30 cm, radius (r) = 7 cm; Conical cavity: height (h) = 24 cm, radius = 7 cm.
Slant height of cone:
$$l = \sqrt{r^2 + h^2} = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \text{ cm}$$
Total Surface Area of remaining solid:
= CSA of cylinder + Area of top circular base + CSA of cone (inner)
$$= 2\pi r H + \pi r^2 + \pi r l$$
$$= \pi r(2H + r + l)$$
$$= \frac{22}{7} \times 7 \times (2 \times 30 + 7 + 25)$$
$$= 22 \times (60 + 7 + 25)$$
$$= 22 \times 92$$
$$= \boxed{2024 \text{ cm}^2}$$
Source: Chapter 12, Section 12.2 – Surface Area of a Combination of Solids
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Explanation
- What surfaces remain visible: (1) curved surface of the full cylinder (outer), (2) the top circular base of the cylinder (the bottom base now has the cone opening into it, so only the top face is a flat circle), (3) the slant/inner surface of the hollowed cone.
- The bottom base of the cylinder is removed when the cone is hollowed out from it — the cone shares the same base, so no flat base area is added there.
- Slant height must be calculated using Pythagoras: $l = \sqrt{r^2+h^2}$.
- Examiner expects the formula written out clearly, substitution shown, and a neat final answer in cm².