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Discover LudwigThe construction "the cost for" is correct and can be used in written English.
It is typically used to indicate the amount of money that is required or needed for something. Here are a few examples: 1. "The cost for attending the conference includes meals and materials." (Here, the phrase is used to specify what is included in the price of attending the conference.) 2. "We are currently negotiating the cost for renting the space for our event." (In this sentence, the phrase is used to specify the price that needs to be paid in order to rent the space.) 3. "What is the cost for shipping these items to our warehouse?" (In this example, the phrase is used to ask about the price of a specific service.) 4. "The cost for repairing the car was much higher than anticipated." (Here, the phrase is used to indicate the amount of money that was spent on repairing the car.).
Dictionary
the cost for
noun
Amount of money, time, etc. that is required or used.
synonyms
Exact(56)
The cost for parents is frankly unimaginable.
The cost for some was dear.
The cost for my fork, for instance?
The cost for the two?
The cost for this arts mecca?
The cost for that expedition is $765.
"The cost for countries is enormous.
The cost for all three is $255.
The cost for two is $2,890.
The cost for a tour is $16.50.
The cost for us is minimal; the cost for others is great.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com