Water is leaking out of a large barrel at a rate proportional to the square rooot of the depth of the water at that time. If the water level starts at 36 inches and drops to 34 inches in 1 hour. How long will it take for all of the water to drain out of that barrel?

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.

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