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A composite solid is formed by attaching a hemisphere of radius 6 cm to the top of a right circular cylinder of height 10 cm and the same radius. Calculate the exact total surface area of the composite solid, including the base of the cylinder but excluding the base of the hemisphere (which is attached).
Use the formula for the surface area of a cylinder (2πrh + 2πr^2) and the surface area of a hemisphere (2πr^2), and express your answer in terms of π.
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Worked solution
- The composite solid consists of a right circular cylinder with radius 6 cm and height 10 cm, and a hemisphere of radius 6 cm attached to the top of the cylinder. The base of the hemisphere is not exposed since it is attached to the cylinder.
- The total surface area includes: the lateral surface area of the cylinder, the base area of the cylinder (since it is included), and the curved surface area of the hemisphere (since the flat base of the hemisphere is not exposed).
- The lateral surface area of a cylinder is given by 2πrh. Substituting r = 6 cm and h = 10 cm: 2 × π × 6 × 10 = 120π cm².
- The base area of the cylinder is a circle with radius 6 cm. Area = πr² = π × 6² = 36π cm².
- The curved surface area of a hemisphere is 2πr². Substituting r = 6 cm: 2 × π × 6² = 72π cm².
- The flat circular base of the hemisphere is not included because it is attached to the cylinder and not exposed.
- Add the three components: lateral surface area of cylinder (120π), base of cylinder (36π), and curved surface area of hemisphere (72π). Total = 120π + 36π + 72π = 228π cm².
- No further simplification is needed since the answer must be in terms of π.
- Check: 120π (cylinder lateral) + 36π (cylinder base) + 72π (hemisphere curved) = 228π cm². All components are accounted for and units are consistent.
- The total surface area of the composite solid, including the base of the cylinder but excluding the base of the hemisphere, is 228π cm².
Reference answer: 228pi cm^2
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