A bucket in the shape of a frustum of a cone has a height of 36 cm. The diameters of its circular ends are 45 cm and 30 cm. A cylindrical tank has a diameter of 2.1 m and a height of 1.6 m. How many b

The formula for the volume of a frustum is $V = \frac{1}{3} \pi h (R^2 + Rr + r^2)$.
From the diagram and problem description:
Height of bucket, $h = 36$ cm
Larger diameter = 45 cm, so larger radius $R = \frac{45}{2} = 22.5$ cm
Smaller diameter = 30 cm, so smaller radius $r = \frac{30}{2} = 15$ cm

Substitute the values into the formula:

$$V_{bucket} = \frac{1}{3} \pi (36) ((22.5)^2 + (22.5)(15) + (15)^2)$$ $$V_{bucket} = 12 \pi (506.25 + 337.5 + 225)$$ $$V_{bucket} = 12 \pi (1068.75)$$ $$V_{bucket} = 12825 \pi \text{ cm}^3$$

The formula for the volume of a cylinder is $V = \pi r^2 h$.
From the problem description:
Diameter of tank = 2.1 m, so radius $r_{tank} = \frac{2.1}{2} = 1.05$ m $= 105$ cm
Height of tank = 1.6 m $= 160$ cm

Substitute the values into the formula:

$$V_{tank} = \pi (105)^2 (160)$$ $$V_{tank} = \pi (11025) (160)$$ $$V_{tank} = 1764000 \pi \text{ cm}^3$$

To find the number of buckets, divide the capacity of the tank by the capacity of the bucket.

$$\text{Number of buckets} = \frac{V_{tank}}{V_{bucket}}$$ $$\text{Number of buckets} = \frac{1764000 \pi \text{ cm}^3}{12825 \pi \text{ cm}^3}$$ $$\text{Number of buckets} = \frac{1764000}{12825}$$ $$\text{Number of buckets} \approx 137.54386$$

Since you cannot draw a fraction of a bucket, and the tank must be filled, we round up to the next whole number.

a)

i) The capacity of the bucket: $\boxed{\text{12825}\pi \text{ cm}^3}$

ii) The capacity of the tank: $\boxed{\text{1764000}\pi \text{ cm}^3}$

b) The number of buckets that must be drawn in order to fill the tank: $\boxed{\text{138 buckets}}$