Okay so someone is probably going to slap me for this answer, but I use jQuery to solve all my irritating problems and it turns out that I just used something today to fix a similar issue. Assuming you use jquery:
$("#content").sibling("#backgroundContainer").css("height",$("#content").outerHeight());
this is untested but I think you can see the concept here. Basically after it is loaded, you can get the height (outerHeight includes padding + borders, innerHeight for the content only). Hope that helps.
Here is how you bind it to the window resize event:
$(window).resize(function() {
$("#content").sibling("#backgroundContainer").css("height",$("#content").outerHeight());
});