In the given figure, a metal container is in the form of a cylinder surmounted by a hemisphere. The internal height of the cylinder is 7 m and the internal radius is 3.5 m. Calculate:

(i) the total area of the internal surface, excluding the base.

(ii) the internal volume of the container in m³.

Given,

Radius of cylindrical portion = Radius of hemispherical portion = r = 3.5 m

Height of cylinder, h = 7 m

(i) Area of internal surface = Surface area of cylinder + Surface area of hemisphere

= 2πrh + 2πr²

= 2πr(h + r)

= 2 × $\dfrac{22}{7}$ × 3.5(7 + 3.5)

= 2 × $\dfrac{22}{7}$ × 36.75

= 2 × 22 × 5.25

= 231 m²

Hence, the total area of the internal surface, excluding the base is 231 m².

(ii) Internal volume of container = Volume of hemisphere + Volume of cylinder

$= \dfrac{2}{3} π\text{r}^3 + π\text{r}^2\text{h} \\[1em] = \dfrac{2}{3} \times \dfrac{22}{7} \times 3.5^3 + \dfrac{22}{7} \times 3.5^2 \times 7 \\[1em] = \dfrac{2}{3} \times \dfrac{22}{7} \times 42.875 + \dfrac{22}{7} \times 12.25 \times 7 \\[1em] = \dfrac{2}{3} \times 22 \times 6.125 + 269.5 \\[1em] = 89.83 + 269.5 \\[1em] = 359.33 \\[1em] = 359 \dfrac{1}{3} \text{m}^3$

Hence, the volume of container is 359.33 m³.