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The phrase "achieved at a cost" is correct and usable in written English.
It can be used to indicate that something has been accomplished but required significant effort, resources, or sacrifices.
Example: "The project was completed successfully, but it was achieved at a cost, as many team members worked overtime to meet the deadline."
Alternatives: "accomplished with sacrifices" or "realized with significant effort".
Exact(13)
All was achieved at a cost of £515.95 ($753).In other cases Mr Sharman formed companies by providing no more than a scanned copy of his driving licence.
This was achieved at a cost of $2,000 per student.These results are only from the first year, but seem to show a clear impact on students attainment in maths.
However, this performance gain is achieved at a cost of no postfabrication flexibility and high engineering design cost.
However, a better PSD of conventional OFDM is achieved at a cost of higher PAPR and BER, as discussed above.
Dundee's midweek derby win, their first defeat of Dundee United in more than a decade, looked to have been achieved at a cost.
And 84% of the reduction can be achieved at a cost lower than 50€/MWh (which is also the natural gas price considered in this scenario).
Similar(47)
This was achieved at a cost-effectiveness ratio of crudely €40,000 per LYG or QALY.
Order is what elephants (that is, Frenchmen) achieve at a cost and with effort.
However, with district heating these goals can be achieved at a lower cost, with heating and cooling costs reduced by approximately 15%.
A net cost over 20 years of $0 was achieved at a private cost of $87 per AMC (excluding 10% program overhead).
Applying a cost function, the optimal number of subjects and time periods are found in such a way that a sufficient power level is achieved at a minimal cost or the power level is maximized for a fixed budget.
More suggestions(16)
provided at a cost
achieved at a price
attained at a cost
delivered at a cost
success at a cost
gain at a cost
completion at a cost
achieved at a velocity
achieved at a point
achieved at a growth
achieved at a strain
achieved at a stroke
achieved at a bias
achieved at a rubber
achieved at a receiver
achieved at a distance
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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