A bucket in the shape of a frustum of a cone has top diameter 40 cm, bottom diameter 30 cm and height 30 cm. A cylindrical tank of radius 1.2 m and height 1.35 m is initially three-fifths full of wate

The bucket is a frustum with:

• Top diameter $D = 40$ cm, so top radius $R = \frac{40}{2} = 20$ cm.
• Bottom diameter $d = 30$ cm, so bottom radius $r = \frac{30}{2} = 15$ cm.
• Height $h = 30$ cm.

The cylindrical tank has:

• Radius $R_{tank} = 1.2$ m $= 1.2 \times 100 = 120$ cm.
• Height $H_{tank} = 1.35$ m $= 1.35 \times 100 = 135$ cm.
• It is initially three-fifths full of water.

a) (i) Calculate the capacity of the bucket in litres, leaving $\pi$ in your answer.

The formula for the volume of a frustum is $V = \frac{1}{3}\pi h (R^2 + Rr + r^2)$.

$$V_{bucket} = \frac{1}{3}\pi (30 \text{ cm}) ((20 \text{ cm})^2 + (20 \text{ cm})(15 \text{ cm}) + (15 \text{ cm})^2)$$ $$V_{bucket} = 10\pi (400 \text{ cm}^2 + 300 \text{ cm}^2 + 225 \text{ cm}^2)$$ $$V_{bucket} = 10\pi (925 \text{ cm}^2)$$ $$V_{bucket} = 9250\pi \text{ cm}^3$$

Since $1 \text{ litre} = 1000 \text{ cm}^3$:

$$V_{bucket} = \frac{9250\pi}{1000} \text{ L}$$ $$V_{bucket} = 9.25\pi \text{ L}$$

The capacity of the bucket is $\boxed{9.25\pi \text{ L}}$.

a) (ii) Calculate the volume of water required to fill the tank in litres.

The formula for the volume of a cylinder is $V = \pi R^2 H$.

$$V_{tank} = \pi (120 \text{ cm})^2 (135 \text{ cm})$$ $$V_{tank} = \pi (14400 \text{ cm}^2) (135 \text{ cm})$$ $$V_{tank} = 1944000\pi \text{ cm}^3$$

The tank is initially three-fifths full, so the remaining volume to fill is two-fifths of the total volume.

$$V_{required} = \frac{2}{5} V_{tank}$$ $$V_{required} = \frac{2}{5} (1944000\pi \text{ cm}^3)$$ $$V_{required} = 777600\pi \text{ cm}^3$$

$$V_{required} = \frac{777600\pi}{1000} \text{ L}$$ $$V_{required} = 777.6\pi \text{ L}$$

The volume of water required to fill the tank is $\boxed{777.6\pi \text{ L}}$.

b) Calculate the number of buckets that must be drawn to fill the tank.

Divide the volume of water required by the capacity of one bucket.

$$\text{Number of buckets} = \frac{V_{required}}{V_{bucket}}$$ $$\text{Number of buckets} = \frac{777.6\pi \text{ L}}{9.25\pi \text{ L}}$$ $$\text{Number of buckets} = \frac{777.6}{9.25}$$ $$\text{Number of buckets} \approx 84.06486$$

Since you cannot use a fraction of a bucket, round up to the nearest whole number.

$$\text{Number of buckets} = 85$$

The number of buckets that must be drawn to fill the tank is $\boxed{85}$.