A model of a cylindrical water tank has a radius of 10.5 cm and a height of 15 cm. The actual water tank has a radius of 210 cm. a) Calculate the volume of the actual water tank. b) Calculate the volu

The linear scale factor is the ratio of corresponding lengths.

$$k = \frac{\text{Actual radius}}{\text{Model radius}} = \frac{r_a}{r_m} = \frac{210 \text{ cm}}{10.5 \text{ cm}} = 20$$

Since the tanks are similar, the ratio of heights is equal to the linear scale factor.

$$h_a = k \times h_m = 20 \times 15 \text{ cm} = 300 \text{ cm}$$

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

$$V_a = \pi r_a^2 h_a$$ $$V_a = 3.142 \times (210 \text{ cm})^2 \times 300 \text{ cm}$$ $$V_a = 3.142 \times 44100 \text{ cm}^2 \times 300 \text{ cm}$$ $$V_a = 3.142 \times 13230000 \text{ cm}^3$$ $$V_a = 41569860 \text{ cm}^3$$

The volume of the actual water tank is $\boxed{41569860 \text{ cm}^3}$.

The volume scale factor is the cube of the linear scale factor.

$$\text{Volume scale factor} = k^3$$ $$\text{Volume scale factor} = (20)^3$$ $$\text{Volume scale factor} = 8000$$

The volume scale factor is $\boxed{8000}$.