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A necklace is made up of wooden beads. Each bead is in the form of a sphere of diameter $4.2 \text{ mm}$. A cylinder is hollowed out from each bead. If the radius of the cylinder is $1 \text{ mm}$, find the volume of wood left in each bead.
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Solution: Radius of a spherical bead $= R = 2.1 \text{ mm}$
radius of cylinder $= r = 1 \text{ mm}$
height of cylinder $= h = 4.2 \text{ mm}$
Volume of wood left in a bead
$= \text{Volume of sphere} - \text{Volume of cylinder}$
$= \frac{4}{3} \pi R^3 - \pi r^2 h$
$= \frac{4}{3} \times \frac{22}{7} \times 2.1 \times 2.1 \times 2.1 - \frac{22}{7} \times 1 \times 1 \times 4.2$
$= 25.608 \text{ cu. mm}$
radius of cylinder $= r = 1 \text{ mm}$
height of cylinder $= h = 4.2 \text{ mm}$
Volume of wood left in a bead
$= \text{Volume of sphere} - \text{Volume of cylinder}$
$= \frac{4}{3} \pi R^3 - \pi r^2 h$
$= \frac{4}{3} \times \frac{22}{7} \times 2.1 \times 2.1 \times 2.1 - \frac{22}{7} \times 1 \times 1 \times 4.2$
$= 25.608 \text{ cu. mm}$