the government are covering up the extent of the continuing disaster. (I especially would like to know the meaning of extent in this sentence) 是什么意思？

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In this case, "extent" refers to the large scale of the disaster. Extent basically means how far something reaches or the degree, significance, or magnitude of something.

So it's basically saying the government is covering up (keeping secret) how big the disaster really is.

Note: It should be "the government is" instead of "the government are" unless there is something plural that comes before it like "members of the government are"