ANS:D - All of these.

1. Half of the top width must be equal to one of the sloping sides: This statement is related to the concept of an optimal trapezoidal channel design. In an economical trapezoidal channel, the top width and the side slopes are designed in a way that they maximize the hydraulic radius for a given cross-sectional area. When half of the top width is equal to one of the sloping sides, it results in a balanced design that minimizes material usage while maintaining hydraulic efficiency.
2. The hydraulic mean depth must be equal to half the depth of flow: This condition does not necessarily apply to the most economical trapezoidal channel design. The hydraulic mean depth is defined as the cross-sectional area divided by the top width. In some cases, the hydraulic mean depth might be close to half the depth of flow, but it's not a strict requirement for an economical trapezoidal channel.
3. The semicircle drawn with the top width as diameter must touch the three sides of the channel: This condition describes the optimal shape for the top of the trapezoidal channel. By drawing a semicircle with the top width as the diameter, the resulting shape ensures that the hydraulic radius is maximized, thus reducing frictional losses. While this condition contributes to the efficiency of the channel, it's not the only factor determining the economy of the design.