Four-stroke cycle). The compression ratio of the engine is 10:1 and the maximum temperature in the cycle is 2000°C. The specific heat ratio (γ) of the working fluid is 1.4. Determine the following:

a) The cut-off ratio
b) The maximum pressure in the cycle
c) The heat rejection during the exhaust stroke
d) The thermal efficiency of the cycle

To solve this problem, we can use the following equations for the Otto cycle:

1) Cut-off ratio (r_c):
r_c = V_3 / V_2

2) Maximum pressure (P_max):
P_max = P_3 = P_4

3) Heat rejection during exhaust stroke (Q_reject):
Q_reject = C_v * (T_4 - T_1)

4) Thermal efficiency (η):
η = 1 - (1 / r_c^(γ-1))

Where:
V_1 = Volume at the beginning of the compression stroke
V_2 = Volume at the end of the compression stroke (beginning of the power stroke)
V_3 = Volume at the end of the power stroke (beginning of the exhaust stroke)
V_4 = Volume at the end of the exhaust stroke (beginning of the intake stroke)
P_1 = Pressure at the beginning of the compression stroke
P_2 = Pressure at the end of the compression stroke (beginning of the power stroke)
P_3 = Pressure at the end of the power stroke (beginning of the exhaust stroke)
P_4 = Pressure at the end of the exhaust stroke (beginning of the intake stroke)
T_1 = Temperature at the beginning of the compression stroke
T_2 = Temperature at the end of the compression stroke (beginning of the power stroke)
T_3 = Temperature at the end of the power stroke (beginning of the exhaust stroke)
T_4 = Temperature at the end of the exhaust stroke (beginning of the intake stroke)
C_v = Specific heat at constant volume

Now let's calculate the values:

a) The cut-off ratio:
r_c = V_3 / V_2
Since it is an air standard cycle, the volumes are related to the compression ratio (r) as follows:
r = V_1 / V_2
r_c = r^(1/γ)
r_c = (10)^(1/1.4)
r_c ≈ 4.6416

b) The maximum pressure in the cycle:
P_max = P_3 = P_4
Since it is an air standard cycle, the pressures are related to the compression ratio (r) as follows:
r = (V_1 / V_2) = (P_1 / P_2)^(1/γ)
P_max = P_1 / r^(γ)
P_max = P_1 / (10)^(1/1.4)

c) The heat rejection during the exhaust stroke:
Q_reject = C_v * (T_4 - T_1)
Since it is an air standard cycle, the temperatures are related to the compression ratio (r) as follows:
r = (V_1 / V_2) = (T_1 / T_2)^(γ-1)
T_4 = T_1 / r^(γ-1)
Q_reject = C_v *