A rectangular tank measuring 40 cm by 30 cm at the base is filled with water at a rate of 1500 cm³/min for 20 minutes. If 1200 cm³ of water overflows, what is the height of the tank?

The water flowed at a rate of $1500 \text{ cm}^3$ per minute for 20 minutes.

$$\text{Total volume flowed} = \text{Flow rate} \times \text{Time}$$ $$\text{Total volume flowed} = 1500 \text{ cm}^3/\text{min} \times 20 \text{ min}$$ $$\text{Total volume flowed} = 30000 \text{ cm}^3$$

The total volume of water flowed into the tank was $30000 \text{ cm}^3$, and $1200 \text{ cm}^3$ of water overflowed. The volume of the tank is the total water flowed minus the overflowed water.

$$\text{Volume of tank} = \text{Total volume flowed} - \text{Volume overflowed}$$ $$\text{Volume of tank} = 30000 \text{ cm}^3 - 1200 \text{ cm}^3$$ $$\text{Volume of tank} = 28800 \text{ cm}^3$$

The base of the rectangular tank measures $40 \text{ cm}$ by $30 \text{ cm}$. The volume of a rectangular tank is given by the formula:

$$\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}$$

We can rearrange this to find the height:

$$\text{Height} = \frac{\text{Volume}}{\text{Length} \times \text{Width}}$$ $$\text{Height} = \frac{28800 \text{ cm}^3}{40 \text{ cm} \times 30 \text{ cm}}$$ $$\text{Height} = \frac{28800 \text{ cm}^3}{1200 \text{ cm}^2}$$ $$\text{Height} = 24 \text{ cm}$$

The height of the tank is $\text{24 cm}$.