to build, to manufacture, to produce something
Are these synonyms, and what is the appropriate context for each of them? What would be appropriate in context of a complex product built in small series?
These are basically synonyms. Produce is the most general of these terms. Production could be willful and intentional as when producing boats, or passive as when producing an odor. It could refer to the inception of a new thing as in movie production, or showing something that already existed, as in "He produced a credit card from his wallet." That last usage is perhaps takes a little artistic license, but it is fairly common.
Build is more specific. It refers to making something new. You can build structures, relationships, organizations, machines, or any number of other things, but generally, it is an intentional act which involves some effort.
In this similar context, manufacture is an even more specific word which refers to a more streamlined and perhaps automated type of building. One thinks of factories where manufacturing takes place. Another meaning refers to something being fake, as in "He manufactured some lame excuse." I suppose that this meaning probably stems from factory manufacture of products which are inferior imitations of their hand-made predecessors.
I would use build or manufacture to talk about boats. If you use the word manufacture, you might draw focus to their lack of individuality, the fact that they are being produced in series. Or it might stress the ease with which several can be produced. If you choose the word build instead, you might draw attention to the effort put into each individual boat, or at the very least avoid the connotations of manufacture. I would not use the word produce. It's so general that it seems empty or overly vague for that context.
The term build suggests the assembly of something from parts, whether those parts are wood or steel (boats) or scraps of code (software).
The term manufacture indicates the creation of some tangible thing using a mechanical or chemical process, regardless of the particular technique, but often on a large scale. It could include boats, steel, or processed food (but not simply grown food).
Produce refers to the creation of any product that might be used or consumed, including manufactured goods (boats, steel, etc.), built goods (boats, software) and harvested goods (such as vegetables, meat, seafood).
There is a lot of overlap in these three words, each of which carries its own special nuances. I have heard of motor cars being 'produced', being 'manufactured' and being 'built'. But generally I would have said that the physical size of the end result is an important determinant.
Nuts and bolts are usually 'produced', as are clothes, footwear, most processed food products etc. (Hence the name 'product').
TV sets and motor vehicles are most often 'manufactured' (produced might be more the mot juste if you were a statistician talking about vast quantities of them,) e.g 'Honda production at Swindon has hit an all-time high' ; but, 'Jaguar, Land Rover are a UK manufacturer of volume cars. Either of these - 'production' and 'manufacture' - could be swapped with one another.
It is also possible, though less common, to speak of building cars e.g. Rolls Royce cars are built at Crewe to individual customer order.
When it comes to things as big as ships, they are invariably 'built', or 'constructed'. e.g. 'The Norfolk Broads used to be home to many firms of boat builders, but nowadays small river craft sold in the UK are mostly built (or since we are talking volumes - 'produced') in the Far East'. But: 'Two Royal Navy aircraft carriers are currently 'under construction' on the Clyde'.
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