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Mathematics · Surface Areas and Volumes · NCERT Examples

Example 6

A complete, independently verified solution for NCERT Example 6.

Mathematics · Chapter 12 · NCERT Examples

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NCERT Example

Example 6

glass capacity · hemispherical raised base

A cylindrical glass has inner diameter 5 cm and height 10 cm. Its bottom has a hemispherical raised portion. Find its apparent and actual capacities. Use π = 3.14.

Cylindrical glass with a hemispherical raised portion at its base.
Figure 12-13 from the supplied NCERT chapter.

Solution

Board-exam working:

r=52=2.5 cm,h=10 cmr=\frac52=2.5\text{ cm},\qquad h=10\text{ cm}

Apparent capacity is the full cylindrical volume.

Va=πr2h=3.14×2.52×10=196.25 cm3V_a=\pi r^2h=3.14\times2.5^2\times10=196.25\text{ cm}^3

The raised hemisphere occupies:

Vh=23πr3=23×3.14×2.53V_h=\frac23\pi r^3=\frac23\times3.14\times2.5^3
≈32.71 cm3\approx32.71\text{ cm}^3

Actual capacity:

V=196.25−32.71=163.54 cm3V=196.25-32.71=163.54\text{ cm}^3

Final answer

Va=196.25 cm3,V=163.54 cm3V_a=196.25\text{ cm}^3,\qquad V=163.54\text{ cm}^3