A solid sphere completely fits into a cylindrical container of radius 3.5 cm and height 7 cm as shown in the figure below. Find the volume of water required to fill the remaining (unshaded) space of the container to the nearest of a whole number. $\left(\pi = \frac{22}{7}\right)$
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Given,
Radius of cylinder (R) = 3.5 cm
Height of cylinder (H) = 7 cm
Since the sphere completely fits into the cylinder,
Radius of sphere ($r$) = 3.5 cm
By formula,
Volume of cylinder = $\pi R^2 H$
$$\begin{aligned}= \frac{22}{7} \times (3.5)^2 \times 7 \\[1em] = \frac{22}{7} \times 12.25 \times 7 \\[1em] = 22 \times 12.25 \\[1em] = 269.5 \text{ cm}^3.\end{aligned}$$
By formula,
Volume of sphere = $\frac{4}{3} \pi r^3$
$$\begin{aligned}= \frac{4}{3} \times \frac{22}{7} \times (3.5)^3 \\[1em] = \frac{4}{3} \times \frac{22}{7} \times 42.875 \\[1em] = \frac{4312}{24} \\[1em] = 179.67 \text{ cm}^3.\end{aligned}$$
Volume of water required = Volume of cylinder − Volume of sphere
= 269.5 − 179.67
= 89.83
≈ 90 cm3.
Hence, the volume of water required = 90 cm3.