A toy is in the form of a cone mounted on a hemisphere of radius 7 cm. The total height of the toy is 31 cm. Find the total surface area of the toy.
Generated by claude-sonnet-4-6 · 2026-06-15 10:35 · grounding rag
Model Answer
Given: radius of hemisphere = radius of cone = r = 7 cm, total height = 31 cm
Height of cone, h = 31 − 7 = 24 cm
Slant height of cone: $l = \sqrt{r^2 + h^2} = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$ cm
Total surface area of toy = CSA of hemisphere + CSA of cone
$$= 2\pi r^2 + \pi r l$$
$$= \pi r(2r + l) = \frac{22}{7} \times 7 \times (14 + 25)$$
$$= 22 \times 39 = \mathbf{858 \ cm^2}$$
Source: Chapter 12, Section 12.2
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Explanation
- The toy's surface consists of only the curved surface of the hemisphere and the curved surface of the cone — the flat circular base is hidden where they join, so it is not included.
- Key step: height of cone = total height − radius of hemisphere (since the hemisphere's height equals its radius).
- Examiner expects the slant height calculation clearly shown, then the final formula $\pi r(2r + l)$ applied correctly.