To calculate the rate of heat transfer through the wall, we need to use the equation for steady-state heat conduction through composite walls:

Q = (T1 - T2) / (Rtotal)

Where:
Q is the rate of heat transfer
T1 and T2 are the temperatures at the two ends of the wall
Rtotal is the total thermal resistance of the wall

The thermal resistance of each layer can be calculated using the formula:

R = d / (k * A)

Where:
R is the thermal resistance of the layer
d is the thickness of the layer
k is the thermal conductivity of the layer
A is the cross-sectional area of the wall

Let's calculate the thermal resistance of each layer first:

R1 = 0.02 m / (10 W/m·K * A)
R2 = 0.05 m / (20 W/m·K * A)
R3 = 0.03 m / (15 W/m·K * A)

The total thermal resistance is the sum of the individual resistances:

Rtotal = R1 + R2 + R3

Now, let's substitute the given values into the equations and calculate the heat transfer rate:

Q = (T1 - T2) / (Rtotal)
= (100°C - 50°C) / (R1 + R2 + R3)

First, we need to calculate the cross-sectional area A. Since we don't have any other information, let's assume the wall has a unit width (1 m):

A = 1 m * 1 m
= 1 m^2

Now, let's calculate the thermal resistances:

R1 = 0.02 m / (10 W/m·K * 1 m^2)
= 0.002 K·m^2/W

R2 = 0.05 m / (20 W/m·K * 1 m^2)
= 0.0025 K·m^2/W

R3 = 0.03 m / (15 W/m·K * 1 m^2)
= 0.002 K·m^2/W

Finally, substitute the values into the equation:

Q = (100°C - 50°C) / (0.002 K·m^2/W + 0.0025 K·m^2/W + 0.002 K·m^2/W)
= 50°C / 0.0065 K·m^2/W
= 7692.31 W

Therefore, the rate of heat transfer through the composite wall is approximately 7692.31 W.

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