A water tank is full up to 8m deep. A drain at the bottom is opened. After 10 minutes of draining, the water is now only 5m deep. Much much later, the tank is (almost) empty.
Assuming exponential decay...

a) Write an equation describing depth as a function of time, treating t = 0 as the time the drain is opened.

b) Predict how much water (as a depth) remains only 3 minutes after the drain is opened.

c) Determine how long it will take for there to be only 1cm of water left in the tank.

d) Find the half-life and time constant for this decay.

e) Use the half-life and/or time constant to plot many data points on time-versus-depth axes and then connect them to graph this function.






A tank of water is already full up to the 30cm mark (with plenty of room for more water). A pump at the bottom is turned on, filling it at an exponentially decaying rate. 23 minutes later, the depth is 53cm, and after a long time, the depth gradually approaches 54 cm.

a) Write an equation describing depth as a function of time, treating t = 0 as the time the pump is turned on. (Be careful: the asymptote is not zero!)

b) Predict the depth of the water at t = 12 minutes.

c) Determine at what time the depth will be 47cm.

d) Find the half-life and time constant for this decay.

e) Use the half-life and/or time constant to plot many data points on time-versus-depth axes and then connect them to graph this function.