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"projected to cost" is correct and usable in written English.
You can use it when you want to express that the estimated cost of something is expected to be a certain amount. For example: "The new bridge is projected to cost around $500 million."
Exact(59)
The tax credits are projected to cost at least $110 billion.
It is projected to cost $240.8 million in 2003 and $260 million next year.
Rebuilding Afghanistan altogether is projected to cost between $14 and $18 billion over the next decade.
So far the renovation has cost $107 million from a campaign projected to cost $200 million.
Despite its size it is projected to cost less than £150 a year to heat.
The plan was projected to cost at least seventy-five million dollars.
The latter is projected to cost $55bn a year for the next two years.
The election is projected to cost $18.5 million, up from $17.1 million in 2004.
The permanent adjustment is projected to cost Medicare $250 billion more over 10 years.
Some season tickets are projected to cost £1,000 more in 2015 than in 2011.
Projected to cost well over $1 billion, it works and must be completed.
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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