To calculate the critical depth of a rectangular channel, we can use the Manning's equation. The critical depth occurs when the specific energy is minimized for a given discharge. The specific energy is the sum of the depth of flow, y, and the velocity head, V^2/2g, where V is the velocity of flow and g is the acceleration due to gravity.

Given:
Width of the rectangular channel (B) = 3 m
Discharge (Q) = 15 m^3/s

1. Calculate the hydraulic radius (Rh):
The hydraulic radius is the cross-sectional area divided by the wetted perimeter. For a rectangular channel, the cross-sectional area (A) is equal to the product of the width (B) and the depth (y), and the wetted perimeter (P) is equal to the sum of the width (B) and twice the depth (y).

A = B * y
P = B + 2y

The hydraulic radius can be calculated as:
Rh = A / P
= (B * y) / (B + 2y)

2. Calculate the velocity of flow (V):
The velocity of flow can be calculated using the discharge (Q) and the cross-sectional area (A):
Q = A * V
V = Q / A

3. Calculate the specific energy (E):
The specific energy is the sum of the depth of flow (y) and the velocity head (V^2/2g):
E = y + (V^2 / 2g)

4. Calculate the critical depth (yc):
To find the critical depth, we need to minimize the specific energy with respect to the depth of flow (y). This can be done by differentiating the specific energy equation with respect to y, setting it equal to zero, and solving for yc.

dE/dy = 1 - (V^2 / gy^2) = 0
V^2 = gy^2
y^3 = V^2 / g
yc^3 = V^2 / g
yc = (V^2 / g)^(1/3)

Substituting the value of V^2 from step 2:
yc = [(Q / A)^2 / g]^(1/3)

Substituting the values of A and P from step 1:
yc = [(Q / (B * y))^2 / g]^(1/3)
= (Q^2 / (B^2 * y^2 * g))^(1/3)
= (Q^2 / (B^2 * g))^(1/3) * (1 / y^(2/3))

5. Calculate the critical depth yc using the given values:
yc = (15^2 / (3^2 * 9.81))^(1/3) * (1 / y^(2/3))

Simplifying the expression:
yc = (225 / 88.29)^(1/3) * (1 / y^(2/3))
= 1.359 * (1 / y^(2/3))

Therefore, the critical depth of the rectangular channel is approximately 1.36 m. Hence, option B is the correct answer.