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Chapter 12 · Question 1

A toy is in the form of a cone of radius 3.53.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.515.5 cm. Find the total surface area of the toy. (Take π=227\pi = \frac{22}{7})

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Q1

A toy is in the form of a cone of radius 3.53.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.515.5 cm. Find the total surface area of the toy. (Take π=227\pi = \frac{22}{7})

Answer Revealed
Direct Answer:
Let the common radius be r=3.5r = 3.5 cm. The height of the hemispherical part is equal to its radius, so height of the hemisphere =r=3.5= r = 3.5 cm. Height of the conical part h=Total height−r=15.5−3.5=12h = \text{Total height} - r = 15.5 - 3.5 = 12 cm. Slant height of the cone l=r2+h2=(3.5)2+122=12.25+144=156.25=12.5l = \sqrt{r^2 + h^2} = \sqrt{(3.5)^2 + 12^2} = \sqrt{12.25 + 144} = \sqrt{156.25} = 12.5 cm. Total surface area of the toy =CSA of cone+CSA of hemisphere=πrl+2πr2=πr(l+2r)=227×3.5×(12.5+2×3.5)=227×72×(12.5+7)=11×19.5=214.5= \text{CSA of cone} + \text{CSA of hemisphere} = \pi r l + 2\pi r^2 = \pi r(l + 2r) = \frac{22}{7} \times 3.5 \times (12.5 + 2 \times 3.5) = \frac{22}{7} \times \frac{7}{2} \times (12.5 + 7) = 11 \times 19.5 = 214.5 cm2^2.

Simple Explanation

The toy is a cone sitting on top of a hemisphere, both with a radius of 3.53.5 cm. The hemisphere takes up 3.53.5 cm of the total 15.515.5 cm height, leaving 1212 cm for the cone. To find the cone's curved surface area, we first need its slant height: l=r2+h2=3.52+122=156.25=12.5l = \sqrt{r^2 + h^2} = \sqrt{3.5^2 + 12^2} = \sqrt{156.25} = 12.5 cm. Now add the curved surfaces: πrl+2πr2=π×3.5×12.5+2π×3.52\pi r l + 2\pi r^2 = \pi \times 3.5 \times 12.5 + 2\pi \times 3.5^2. Using π=227\pi = \frac{22}{7} and simplifying gives 11×19.5=214.511 \times 19.5 = 214.5 cm2^2. The flat circular face between the cone and hemisphere is hidden inside and does not count towards the surface area.

Exam-Ready Structure

This problem tests the surface area of a solid formed by combining a cone and a hemisphere: 1. Identify dimensions: Radius of both cone and hemisphere r=3.5r = 3.5 cm. Total height H=15.5H = 15.5 cm. 2. Since the hemisphere sits below the cone with its flat face upward, the height of the hemispherical part equals its radius r=3.5r = 3.5 cm. Therefore, height of conical part h=H−r=15.5−3.5=12h = H - r = 15.5 - 3.5 = 12 cm. 3. Slant height of the cone is essential because the curved surface area formula requires it: l=r2+h2=(3.5)2+122=12.25+144=156.25=12.5l = \sqrt{r^2 + h^2} = \sqrt{(3.5)^2 + 12^2} = \sqrt{12.25 + 144} = \sqrt{156.25} = 12.5 cm. 4. The total surface area of the combined solid == CSA of cone ++ CSA of hemisphere. The flat circular faces that are joined together are not part of the external surface. TSA =πrl+2πr2=πr(l+2r)= \pi r l + 2\pi r^2 = \pi r(l + 2r). 5. Substitute π=227\pi = \frac{22}{7}: TSA =227×3.5×(12.5+7.0)=227×72×19.5=11×19.5=214.5= \frac{22}{7} \times 3.5 \times (12.5 + 7.0) = \frac{22}{7} \times \frac{7}{2} \times 19.5 = 11 \times 19.5 = 214.5 cm2^2. 6. Key principle: When two solids are joined along a common face, that common face is not part of the outer surface. Only the exposed (visible) surfaces contribute to the total surface area. This is different from finding the volume, where volumes of constituents are simply added.

Key Points

  • Common radius r=3.5r = 3.5 cm for both cone and hemisphere
  • Height of hemisphere equals its radius: r=3.5r = 3.5 cm
  • Height of cone h=15.5−3.5=12h = 15.5 - 3.5 = 12 cm; slant height l=r2+h2=12.5l = \sqrt{r^2 + h^2} = 12.5 cm
  • TSA =πrl+2πr2= \pi r l + 2\pi r^2 (the joined flat faces are not part of the surface)
  • Final answer: 214.5214.5 cm2^2

Common Mistakes

  • Adding TSA of cone (which includes base area) instead of only CSA of cone — the base of the cone is hidden inside the joint, so only its curved surface is visible
  • Using total height directly as cone height without subtracting the hemispherical radius

Related Questions

Q2

Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no colour on it. He wanted to colour it with his crayons. The top is shaped like a cone surmounted by a hemisphere. The entire top is 55 cm in height and the diameter of the top is 3.53.5 cm. Find the area he has to colour. (Take π=227\pi = \frac{22}{7})

Q3

A wooden toy rocket is in the shape of a cone mounted on a cylinder. The height of the entire rocket is 2626 cm, while the height of the conical part is 66 cm. The base of the conical portion has a diameter of 55 cm, while the base diameter of the cylindrical portion is 33 cm. If the conical portion is to be painted orange and the cylindrical portion yellow, find the area of the rocket painted with each of these colours. (Take π=3.14\pi = 3.14)

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