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A DC electric motor is connected to a wheel via a gearbox. The aim is to find when to engage
the gear so that the torque at the wheel is always the largest possible.
On its own, the motor generates a torque 𝑀 (Ω) (in Nm) that depends on the rotation speed Ω
(in rpm) of the motor shaft:
𝑀 (Ω) = 0.3 (1 − Ω/400)

The torque 𝑇 (𝜔) exerted at the wheel depends on the rotation speed 𝜔 of the wheel, and on
whether the gear is engaged:
𝑇0 (𝜔) = 𝑀 (𝜔), if the gear is not engaged,
𝑇1 (𝜔) = 1/3 𝑀  𝜔/3
, if the gear is engaged.
(a) Write explicit expressions for 𝑇0 (𝜔) and 𝑇1 (𝜔) as functions of 𝜔 given the definition of
𝑀 (Ω).
(b) Plot the two functions you found in (a) on a same graph for positive values of 𝜔.
(c) Find the optimal gear shifting point 𝜔∗ that ensures that the torque 𝑇 (𝜔) is always as
large as possible.
Hint: Look for a graphical solution first by using the graph from Part (b).