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Answer:
It will take about 35.49 hours for the water to leak out of the barrel.
Step-by-step explanation:
Let be the depth of water in the barrel at time
, where
is measured in inches and
in hours.
We know that water is leaking out of a large barrel at a rate proportional to the square root of the depth of the water at that time. We then have that
where is a constant of proportionality.
Separation of variables is a common method for solving differential equations. To solve the above differential equation you must:
Multiply by
Multiply by
Take integral
Integrate
Isolate
We know that the water level starts at 36, this means . We use this information to find the value of
.
At t = 1, y = 34
So our formula for the depth of water in the barrel is
To find the time, , at which all the water leaks out of the barrel, we solve the equation
Thus, it will take about 35.49 hours for the water to leak out of the barrel.