I highly doubt a Task Killer would kill the battery THAT quickly as I too ran one when I first got my mytouch4g and I can still go a day of moderate use without killing my battery.
I also doubt it's the free app you downloaded since I have downloaded every single free app amazon has offered and haven't noticed a thing.
What type of phone do you have? OS? When did you get it? Do you have brightness turned all the way up, gps, wifi, bluetooth enabled?
As I always say (I should just add it to my sig...) the more information you offer the better people can help you. Hope you get it fixed asap. Maybe you spilled something on your phone?
Just want to point out, as far as free apps you may be talking about different things. Amazon has many, many free apps, not just their free app of the day. So he may well have downloaded a free app that's gone rogue. In fact, if it IS an app problem, I'd say it's more likely to come from a free one that a paid one.
I have an HTC Incredible (its listed in my profile) running stock Android 2.2. Not rooted. Bluetooth & GPS are off, but wifi is usually always on. Brightness is always on medium.
This isn't the first time this has happened, some days are better than others. Sometimes when I get home from work the phone still has 75+% left, others its below 50%. In both cases I haven't really used the phone all day. But last night and this morning it was really bad. Phone was warm, too.
Also I don't actually "run" a task manager, I just use it when I want to kill processes that don't have a "Exit" on the menu. Like when I'm trying to troubleshoot whats killing my battery.
Thats the first I've heard of booting into CWM, I'll check it out, thanks.
First things first: killing processes using a task manager is bad. I know you're not "running" it per se, and there's a long, complicated explanation that totally defies conventional computer logic... but killing apps you're not using is bad. Check out the articles listed or do a few searches on your own if you're really curious as to why.
Secondly, it sounds like your battery problem isn't really a "problem," just the result of poor software/app management. There's a billion (dozen) things to do to help with battery life, I won't get into many of them here (since there really are many things) but a few searches on saving battery life should help. A few of the biggies, though, are:
Display brightness. Set your display to auto, or even manually lower it if you can stand it.
GPS/signal connection: Turn GPS off when you're not using it, and anytime you're not getting a signal, switch to airplane mode. If your office has spotty signal strength, your phone may be searching for a signal all day, which will definitely kill your battery.
Updating apps: Apps like facebook, twitter, or weather may be constantly updating with new information-- every two or three minutes perhaps. Check the settings on these to see when they're updating, and make sure they're updating far less frequently. Do you REALLY need weather updates every three minutes? I set mine to two hours and it works pretty well.
Finally, it COULD be a hardware problem. When your battery's getting low, go into settings -> battery management, and see what's there. You can take a screenshot if you know how, or just type out what's there and we can help you. If there's any major apps or anything killing your battery, we'll be able to see it, and if there's no reasonable explanation for terrible battery life, it's probably a hardware problem and something needs to be replaced.
Some additional notes:
Depending on how long you work, going all day and getting home with 75%-50% battery sounds pretty good to me. After I've been at school all day (anywhere from 6-12 hours, depending on my class schedule), I'm generally around 50%-30%. I'd love to do that and have 75% left.
Just one of the cons to having a smartphone these days-- your battery drains faster than when you had your Razr (man, that phone was cool at the time though).
The phone being warm is slight cause for concern. It could be normal, and remember that hotter batteries drain faster, so keep it in a cool place if you can (don't leave it sitting in the sun), but it COULD indicate a hardware problem.