Problem formulation

[vvaann]

The inside of a certain water tank is a cube measuring 10 feet on each edge and having vertical sides and no top. Let h(t) denote the water level, in feet, above the floor of the tank at time t seconds. Starting at time t=0, water pours into the tank at a constant rate of 1 cubic foot per second, and simultaneously, water is removed from the tank at a rate of 0.25h(t) cubic feet per second. As t -> infinity, what is the limit of the volume of the water in the tank?

A. 400 cubic feet
B. 600 cubic feet
C. 1,000 cubic feet
D. The limit does not exist
E. The limit exists, but it cannot be determined without knowing h(0).

Answer key:

SPOILER: A

How do you make an unambiguous formulation of this prolem?

Br,
vvaann

[reactor]

I say A. My thinking is a follows: at h(t)=4 -- 0.25h(t) = 1. Therefore the input and output rates are equal. When this "equilibrium" is reached the tank will be filled 4 x 10 x10 =400 cubic feet. [also keep in mind that for h(t)<4 --0.25h(t) < 1, thus input>output and the water level will indeed reach 4 feets. Also h(0) is obviously 0 since it is stated explicitly in the problem that "Starting at time t=0, water pours into the tank...." ]

I know that the above is not unambiguous but I have never studied for Math GRE. :-)

[vvaann]

Very nice, gstergia!

I think your approache gave me an general indea about how to deal with this kind of problem. Thanks a lot :-)