Rate of Surface Rise in a Rectangular Trough

Consider a rectangular trough with a unique shape characterized by a length of 8 meters, a width of 2 meters at the top, and a depth of 4 meters. Water flows into this trough at a steady rate of 2 cubic meters per minute. Given these dimensions and the flow rate, we can explore the rate at which the surface of the water rises, particularly when the water is 1 meter deep.

Understanding the Geometry of the Rectangular Trough

The first step is to recognize the geometry of the trough. The volume of an empty trough is given by the formula length × width × depth, which for our trough calculates to 8 meters × 2 meters × 4 meters 64 cubic meters (m3).

Water Flow and Volume Calculation

Water flows into the trough at a rate of 2 cubic meters per minute. To determine how long it takes to fill the trough, we can use the total volume of the trough. It would take 64 m3 ÷ 2 m3 / min 32 minutes to completely fill the trough. However, our current interest is when the water is only 1 meter deep.

The Rate of Surface Rise

When the water is 1 meter deep, the volume of water in the trough is 8 meters × 2 meters × 1 meter 16 cubic meters. Given that the water flows in at a rate of 2 m3 per minute, it would take 16 m3 ÷ 2 m3 / min 8 minutes to reach this depth.

Uniform Rate of Surface Rise

Crucially, the rate at which the surface of the water rises is uniform throughout the depth of the water. This is because the horizontal cross-section of the trough does not change with depth. This means that the speed of the water’s surface rise remains constant, regardless of the height of the water. Therefore, when the water is 1 meter deep, the rate of surface rise is 8 meters per 60 minutes, which simplifies to 1/8 meter per minute.

To reiterate, the rate of surface rise is consistent and does not depend on the depth of the water. Regardless of the height, the surface of the water rises at a rate of 1/8 meter per minute, as long as the water is continuously flowing into the trough at the specified rate.

Conclusion

In conclusion, when a rectangular trough with a length of 8 meters, a width of 2 meters at the top, and a depth of 4 meters is filled at a rate of 2 cubic meters per minute, the surface of the water rises uniformly. When the water is 1 meter deep, it takes 8 minutes to fill, and the surface rises at a rate of 1/8 meter per minute. This uniformity in the rate of surface rise is due to the consistent horizontal cross-section of the trough, ensuring a steady and predictable water level increase.

Finding Water Surface Rise Rates

If you are dealing with similar problems related to the rate of surface rise in various shapes or need to calculate rates of water surface rise in different containers, understanding the geometry and the flow rate is crucial. Whether it is a rectangular trough, a cylindrical tank, or any other shape, the principle remains the same. Knowing the volume flow rate and the dimensions of the container allows for accurate calculations of the rate of surface rise.