(Warm) Air intake hose

Warm air duct conducts heated air from the exhaust manifold 'stove' in an effort to provide a more consistant air intake temperature.
and yes, faster warm ups.
The vacuum diaphragm (aka vacuum motor) usually is controlled by an air temperature sensitive valve in the air cleaner housing.

Described on page 2 of the '74 Cleaner Air System
https://mymopar.com/browse-mtsc-by-model-year-1974/