The diagram below shows a net of a cone. All ABC is a semi-circle of radius 3.5 cm. Find the volume of the cone, by taking the value of pi to be 22/7. What is the difference in the surface area of the hemisphere and the total surface area of the cone?

Rediasi ya hemisifia na koni, $r = 3.5 \text{ cm}$.
Jumla ya urefu wa gimba, $H = 12 \text{ cm}$.
Thamani ya $\pi = \frac{22}{7}$.

Urefu wa hemisifia ni sawa na rediasi yake, yaani $3.5 \text{ cm}$.
Urefu wa koni ($h_{koni}$) utakuwa jumla ya urefu kutoa urefu wa hemisifia:

$$h_{koni} = H - r = 12 \text{ cm} - 3.5 \text{ cm} = 8.5 \text{ cm}$$

Fomula ya ujazo wa hemisifia ni $\frac{2}{3} \pi r^3$.

$$V_{hemisifia} = \frac{2}{3} \times \frac{22}{7} \times (3.5)^3$$ $$V_{hemisifia} = \frac{2}{3} \times \frac{22}{7} \times 42.875$$ $$V_{hemisifia} = \frac{1886.5}{21}$$ $$V_{hemisifia} \approx 89.833 \text{ cm}^3$$

Fomula ya ujazo wa koni ni $\frac{1}{3} \pi r^2 h_{koni}$.

$$V_{koni} = \frac{1}{3} \times \frac{22}{7} \times (3.5)^2 \times 8.5$$ $$V_{koni} = \frac{1}{3} \times \frac{22}{7} \times 12.25 \times 8.5$$ $$V_{koni} = \frac{2289.25}{21}$$ $$V_{koni} \approx 109.012 \text{ cm}^3$$

Ujazo jumla ni jumla ya ujazo wa hemisifia na ujazo wa koni.

$$V_{jumla} = V_{hemisifia} + V_{koni}$$ $$V_{jumla} = 89.833 \text{ cm}^3 + 109.012 \text{ cm}^3$$ $$V_{jumla} = 198.845 \text{ cm}^3$$

Kwa kutumia sehemu kamili kwa usahihi zaidi:

$$V_{hemisifia} = \frac{539}{6} \text{ cm}^3$$ $$V_{koni} = \frac{1309}{12} \text{ cm}^3$$ $$V_{jumla} = \frac{539}{6} + \frac{1309}{12} = \frac{1078}{12} + \frac{1309}{12} = \frac{2387}{12} \text{ cm}^3$$ $$V_{jumla} \approx 198.917 \text{ cm}^3$$

Ujazo wa gimba ni:

$$\text{198.917 cm}^3$$