Final answer:
This would be an example of vehicle-borne transmission, with the contaminated water acting as the vehicle for transmitting the gastrointestinal pathogen to the farmer's family.
Step-by-step explanation:
This would be an example of vehicle-borne transmission. In this case, the vehicle of transmission is the contaminated water from the farmhouse well. The cattle, which are allowed to pasture in the field, have likely contaminated the water with a gastrointestinal pathogen through their feces or another source of contamination. When the farmer's family drinks the contaminated water, they become ill with the pathogen.