Imagine
a 10 gallon bucket that is filled by a faucet.We refer to the faucet as the source of water flowing into the bucket, S.If S=2.0 gallon per minute (2.0 gal/min) then the bucket will become
completely filled in 5 minutes.The
amount of water in the bucket at any given time will be referred to as the
bucket content, C.

If
we start with an empty bucket and turn on the faucet at a constant flow (S in
gallons per minute), then the
water content C at any time t is
easy to calculate from

C = S t

Complete the table below for S=2.0 gal/min (gallons per minute)

QuesA1:

Check
your understanding.If the bucket
starts out empty (C_{0}=0) and is filled at a rate of S=3.0 gal/min, how
much water will be in the bucket in 3.0 minutes? The
symbol C_{0}
stands for the initial bucket content at time t=0.

To help you check your answers, the
graph below shows:

1)
How the bucket drains if the drain rate is a constant 1 gal/min (black line). Note that the bucket completely drains under this assumption in one life-time.

2)
How the bucket drains if the drain rate at any instant is always equal to 10% of
what's in the bucket. (red line) For this exact solution 37% of the water remains in the bucket after one
life-time.

3)
How the bucket drains if we assume the for each 1-minute time interval the drain
rate is constant and equal to 10% of what's in bucket at the start of the
1-minute time interval. (blue diamonds) This is what we did for the above
table. The errors introduced by this approximation are relatively small.