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Mathematics · Surface Areas and Volumes · NCERT Examples

Example 1

A complete, independently verified solution for NCERT Example 1.

Mathematics · Chapter 12 · NCERT Examples

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NCERT Example

Example 1

combined surface area · cone and hemisphere

A playing top is shaped like a cone surmounted by a hemisphere. Its total height is 5 cm and its diameter is 3.5 cm. Find the area to be coloured. Take π = 22/7.

Playing top formed by a hemisphere over a cone, total height 5 centimetres and diameter 3.5 centimetres.
Figure 12-6 from the supplied NCERT chapter.

Solution

Board-exam working:

The common radius is half the diameter.

r=3.52=1.75 cmr=\frac{3.5}{2}=1.75\text{ cm}

Height of the conical part:

h=5−1.75=3.25 cmh=5-1.75=3.25\text{ cm}

Slant height of the cone:

l=r2+h2=1.752+3.252l=\sqrt{r^2+h^2}=\sqrt{1.75^2+3.25^2}
=13.625≈3.7 cm=\sqrt{13.625}\approx3.7\text{ cm}

Only the curved surfaces of the hemisphere and cone are exposed.

A=2πr2+πrl=πr(2r+l)A=2\pi r^2+\pi rl=\pi r(2r+l)
=227×1.75×(3.5+3.7)=\frac{22}{7}\times1.75\times(3.5+3.7)
=39.6 cm2 (approximately)=39.6\text{ cm}^2\text{ (approximately)}

Final answer

39.6 cm2 (approximately)\boxed{39.6\text{ cm}^2\text{ (approximately)}}