The correct answer is $\boxed{\text{A) 2.4 m}}$.
The center of pressure is the point on a surface where the total force of the fluid acting on the surface can be considered to be applied. The depth of the center of pressure can be calculated using the following formula:
$$d = \frac{h}{2} + \frac{b}{3}$$
where $h$ is the height of the gate and $b$ is the width of the gate.
In this case, $h = 6$ m and $b = 8$ m. Substituting these values into the formula, we get:
$$d = \frac{6}{2} + \frac{8}{3} = 2.4\text{ m}$$
Therefore, the depth of the center of pressure is 2.4 m.
Option B is incorrect because it is the depth of the center of pressure for a gate that is 4 m high. Option C is incorrect because it is the depth of the center of pressure for a gate that is 8 m wide. Option D is incorrect because it is the depth of the center of pressure for a gate that is 2.5 m high.