Write an expression for the rate of water being pumped into the tank.

Question

water is pumped into an underground tank at a constant rate of 8 gallons per minute. water leaks out of the tank at the rate of √t+1 gallons per minute, for 0≤t≤120 minutes. At time t=0, the tank contains 30 gallons of water.

Answer 1

Hey Bethany!

I might need a bit of clarification here, because the expression for the rate of water into the tank is actually just 8.

This is a classic tank problem in differential equations, and often times it will even go as far as asking of the concentration of the water as well!

For our purposes, let's come up with a function that we'll call V(t) to represent the volume of water in the tank at any given time, t.

Then we can think of our rate of change of the volume in the tank (dV/dt) as follows:

dV/dt = rate of water in - rate of water out.

Water comes in at a constant rate of 8 gal/min and it goes out at a variable rate of √t+1 gal/min.

So our equation would be:

dV/dt = 8 - (√t+1)

or simplified: dV/dt = 7-√t

From there, questions like this normally will ask you to find an expression for V(t).

You could integrate with respect to t on both sides, and then we could use the fact in the question that

V(0) = 30 to figure out our constant of integration (C).

Hope that helps!