Question

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A tank holds 400 gallons of water. A crack forms and 20 gallons of water leak out every hour. Write and simplify an expression to find how much water is in the tank after 12 hours.

Answer 1

The volume of water in the tank decreases by 20 gallons per minute, so you can set up a pretty simple differential equation relating the change in volume \(V\):
\[\frac{dV}{dt}=-20\]
Solving yields \(V=-20t+C\). You know that at time \(t=0\), you have a volume of 400 gallons (assuming the tank to be full initially). This means you have
\[400=-20(0)+C~~\implies~~C=400\]
and so the volume at any time \(t\ge0\) is
\[V=-20t+400\]