Can nobody help me with these problems?
Hey guys, I need some help with some integration problems. Some I could solve already but there are two where I need help. If you can help me, however, please don't just say me: integrate this, but also show me how to do it, that would be really nice. Thanks a lot for everyone who helps me.
1: A large bottle with, 4m high and with a radius of 2m, loses water. Originally, the bottle was full and lost water at a rate proportional to the square root of the depth of the remaining water.
I'm given that the depth of the water is 1m after 2 hours. The question is after how many hours the tank is empty. Please help me with this question.
2. In a freezing machine the temperature is constantly 5 degree. An 100 degree hot object is put in it. One minute later the object is only 80 degree hot. When will it reach a temperature of 10 degree?
Thanks a lot for every advice. But please also give me explanations if you can. I very much appreciate any help. Thanks a lot.
I have a long, looong solution . . . maybe someone can shorten it.
1) A large bottle, 4m high and with a radius of 2m, loses water.
Originally, the bottle was full and lost water at a rate proportional
to the square root of the depth of the remaining water.
The depth of the water is 1m after 2 hours.
After how many hours the tank is empty?
The volume of a cylinder is: .
Since , the volume of the water is: .
. . Then: .
But we are told that: .
So we have: . . . . a differential equation
. . Separate variables: .
. . Integrate: .
. . We have: .
The equation (so far) is: .
The equation is: .
Hence, the height function is: .
If the tank is empty, then
We have: .
Therefore, it takes 4 hours to empty the tank.
But check my reasoning and my work . . . please!