Amos has 3200 cm³ of liquid paraffin. He poured the paraffin into a cone of radius 10 cm. What is the height of the cone if it was completely filled with paraffin?

$$3200 = \frac{1}{3} \times \pi \times (10)^2 \times h$$ $$3200 = \frac{1}{3} \times \pi \times 100 \times h$$

$$3200 \times 3 = 100 \pi h$$ $$9600 = 100 \pi h$$ $$h = \frac{9600}{100 \pi}$$ $$h = \frac{96}{\pi}$$

$$h \approx \frac{96}{3.14159}$$ $$h \approx 30.557 \text{ cm}$$ $$h \approx 30.56 \text{ cm}$$

Comparing with the options, option A is $30.55 \text{ cm}$.
The height of the cone is $\boxed{\text{30.55 cm}}$.

:
We need to calculate the total percentage loss of chickens.

$$\text{Total chickens} = 1200$$ $$\text{Vaccinated chickens} = 1020$$ $$\text{Unvaccinated chickens} = 1200 - 1020 = 180$$

$$\text{Percentage loss} = \frac{\text{Total died}}{\text{Total chickens}} \times 100\%$$ $$\text{Percentage loss} = \frac{159}{1200} \times 100\%$$ $$\text{Percentage loss} = 0.1325 \times 100\%$$ $$\text{Percentage loss} = 13.25\%$$

The correct option is C.
The percentage loss was $\boxed{\text{13.25\%}}$.

:
The area of a triangle given two sides and the included angle is calculated using the formula: Area $= \frac{1}{2}ab \sin C$.
We are given $a = 50 \text{ m}$, $b = 70 \text{ m}$, and $C = 80^\circ$.

$$\text{Area} = \frac{1}{2} \times 3500 \times \sin(80^\circ)$$ $$\text{Area} = 1750 \times 0.98480775$$ $$\text{Area} \approx 1723.4135 \text{ m}^2$$

$$\text{Area} \approx 1723.41 \text{ m}^2$$

The correct option is A.
The area of the pipeline is $\boxed{\text{1723.41 m}^2}$.

:
The curved surface area of a hemisphere is given by the formula $A = 2\pi r^2$.
We are given the radius $r = 6.5 \text{ cm}$.

$$A = 2 \times \pi \times 42.25$$ $$A = 84.5 \pi$$ $$A \approx 84.5 \times 3.14159$$ $$A \approx 265.464 \text{ cm}^2$$

$$A \approx 265.46 \text{ cm}^2$$

The correct option is D.
The area of the curved hemispherical surface is $\boxed{\text{265.46 cm}^2}$.

:
The volume of a cone is given by the formula $V = \frac{1}{3}\pi r^2 h$.
We are given the radius $r = 3 \text{ cm}$ and the vertical height $h = 10 \text{ cm}$.

$$V = \frac{1}{3} \times \pi \times 9 \times 10$$ $$V = 3 \times \pi \times 10$$ $$V = 30\pi$$ $$V \approx 30 \times 3.14159$$ $$V \approx 94.2477 \text{ cm}^3$$

$$V \approx 94.25 \text{ cm}^3$$

The correct option is A.
The volume of the cups is $\boxed{\text{94.25 cm}^3}$.

:
We need to find the cost of manufacturing the fruit juice. We will work backward from the wholesaler's selling price.

The wholesaler sells for Sh 84, making a 20% profit. This means Sh 84 represents 120% of the price the wholesaler paid.
Let $P_W$ be the price the wholesaler paid.

$$1.20 \times P_W = 84$$ $$P_W = \frac{84}{1.20}$$ $$P_W = 70 \text{ Sh}$$

So, the manufacturer sold the juice to the wholesaler for Sh 70.

The manufacturer sold the juice for Sh 70, making a 40% profit. This means Sh 70 represents 140% of the manufacturing cost.
Let $C_M$ be the cost of manufacturing.

$$1.40 \times C_M = 70$$ $$C_M = \frac{70}{1.40}$$ $$C_M = 50 \text{ Sh}$$

The correct option is D.
The cost of manufacturing the fruit juice is $\boxed{\text{Sh 50}}$.

2 zimekamilika. Zimebaki 3 leo — tuma inayofuata.