A hemispherical and a conical hole is scooped out of a solid wooden cylinder. Find the volume of the remaining solid where the measurements are as follows :

The height of the solid cylinder is 7 cm, radius of each of hemisphere, cone and cylinder is 3 cm. Height of cone is 3 cm. Give your answer correct to the nearest whole number.

Given,

Radius, r = 3 cm

Height of cone, h = 3 cm

Height of cylinder, H = 7 cm

From figure,

Volume of remaining solid = Volume of cylinder - Volume of cone - Volume of hemisphere

∴ Volume of remaining solid = πr²H - $\dfrac{1}{3}$ πr²h - $\dfrac{2}{3}$ πr³

$= π\text{r}^2(\text{H} - \dfrac{1}{3}\text{h} - \dfrac{2}{3} \text{r}) \\[1em] = \dfrac{22}{7} \times 3^2(7 - \dfrac{1}{3} \times 3 - \dfrac{2}{3} \times 3) \\[1em] = \dfrac{22}{7} \times 9(7 - 1 - 2) \\[1em] = \dfrac{22}{7} \times 9 \times 4 \\[1em] = \dfrac{792}{7} \\[1em] = 113.14 \approx 113 \text{ cm}^3.$

Hence, the volume of the remaining solid is 113 cm³.