ausgedehnt
extensive
Describes something that extends over a large spatial area; expansive, widespread.
Hinter dem Haus erstreckt sich ein ausgedehnter Park.
An extensive park stretches out behind the house.

extensive / extended
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extensive
Describes something that extends over a large spatial area; expansive, widespread.
Hinter dem Haus erstreckt sich ein ausgedehnter Park.
An extensive park stretches out behind the house.

'Ausgedehnt' (extensive/extended) is the past participle of the verb 'ausdehnen' (to expand/extend). 'Dehnen' means 'to stretch'. The prefix 'aus-' reinforces this. Literally, it therefore describes something that has been 'stretched out' or expanded, which applies to both space and time.
Die Wüste dehnt sich immer weiter aus.
The desert is expanding farther and farther.
You can tell whether 'ausgedehnt' (extensive/extended) is meant spatially or temporally from the noun. If it refers to a place (e.g., forest, area, beach), it is about area. If it refers to an activity (e.g., breakfast, conversation, trip), it is about duration.
Wir wanderten durch ein ausgedehntes Waldgebiet.
We hiked through an extensive forest area.
In its spatial meaning, 'ausgedehnt' (extensive) is often used to describe large natural areas or landscapes. Typical combinations are 'ausgedehnte Wälder' (extensive forests), 'ein ausgedehntes Gebiet' (an extensive area), or 'ausgedehnte Felder' (extensive fields).
Die Region ist bekannt für ihre ausgedehnten Weinberge.
The region is known for its extensive vineyards.
When it is about time, 'ausgedehnt' (extended) often describes pleasant or intensive activities. Common phrases include 'ein ausgedehntes Frühstück' (an extended breakfast), 'ein ausgedehnter Spaziergang' (an extended walk), or 'eine ausgedehnte Reise' (an extended trip).
Am Sonntag genießen wir immer ein ausgedehntes Frühstück mit der ganzen Familie.
On Sundays we always enjoy an extended breakfast with the whole family.
far / wide
'weit' and 'ausgedehnt' can both describe something as spatially large. 'weit' is more general and fits with distances, paths, views, or landscapes. 'ausgedehnt' is used more when something stretches over a large area, for example a forest, a region, or a network of paths. It sounds somewhat more precise and often more factual than 'weit'. If you only want to say that a destination is not nearby, use 'weit'. If you want to emphasize the large area or extent, 'ausgedehnt' fits better.

Hinter dem Dorf beginnt ein ausgedehntes Waldgebiet.
An extensive forest area begins behind the village.
Von der Burg hat man einen weiten Blick über das Tal.
From the castle, you have a wide view over the valley.