[b][[/b][size=85][size=100]Read either the bold [/size][size=150][color=#0000ff][b][i]blue[/i][/b] [/color][/size][size=100]words or the bold [/size][size=150][b][color=#6aa84f][i]green[/i][/color] [/b][/size][/size]words along with the normal black words[b]][/b][br][br]If [i][b][color=#6aa84f]WATER[/color][/b][/i]/[i][b][color=#0000ff]MONEY[/color][/b][/i] comes into a [color=#6aa84f][i][b]BATHTUB[/b][/i][/color]/[color=#0000ff][i][b]BANK[/b][/i][/color] at a constant rate[br]and leaks out a a rate proportional to the [color=#6aa84f][i][b]HEIGHT[/b][/i][/color]/[color=#0000ff][i][b]AMOUNT[/b][/i][/color] of the[br][i][b][color=#6aa84f]WATER[/color][/b][/i]/[color=#0000ff][i][b]MONEY[/b][/i][/color] in the [color=#6aa84f][i][b]BATHTUB[/b][/i][/color]/[color=#0000ff][i][b]BANK[/b][/i][/color], then an equilibrium [color=#6aa84f][i][b]HEIGHT[/b][/i][/color]/[color=#0000ff][i][b]AMOUNT[/b][/i][/color] of [color=#6aa84f][i][b]WATER[/b][/i][/color]/[color=#0000ff][i][b]MONEY[/b][/i][/color] in the [color=#6aa84f][i][b]BATHTUB[/b][/i][/color]/[color=#0000ff][i][b]BANK[/b][/i][/color] can be reached.[br][br]You can use this applet to explore how the equilibrium depends[br]on the rate of flow into and the rate of flow out of the [color=#6aa84f][i][b]BATHTUB[/b][/i][/color]/[color=#0000ff][i][b]BANK[/b][/i][/color].[br][br]Clearly, the [color=#6aa84f][i][b]BATHTUB[/b][/i][/color]/[color=#0000ff][i][b]BANK[/b][/i][/color] is a metaphor for many equilibrium situations[br]that arise because of competing rates. Can you think of others?[br][br]The applet chooses a random initial value of [color=#6aa84f][i][b]WATER[/b][/i][/color]/[color=#0000ff][i][b]MONEY[/b][/i][/color] in the [color=#6aa84f][i][b]BATHTUB[/b][/i][/color]/[color=#0000ff][i][b]BANK[/b][/i][/color]. You can then watch an animation by clicking on the "play" icon in the lower left hand corner of either window.[br][br][b][i][color=#ff0000]What have you learned about models and modeling while exploring this applet?[/color][/i][/b][br][br][b][color=#ff0000][size=85]Further possible applications are discussed below the screen image.[/size][/color][/b][br][br][color=#0000ff][size=85][*if the bathtub is level and has vertical sides, then the height and volume of water are proportional to one another][/size][/color]

Here is another physical situation can be modeled this way - can you think of others?[color=#ff0000][b][br][br]Suppose you have a helium balloon to which is tied a very long rope. The helium balloon is released with the end of the rope still dragging on the ground.[br][br] - If it released from a large height the weight of the rope it is supporting will pull the balloon down to some equilibrium height[br][br]- if it is released near the ground the balloon will rise because the amount of rope that it is supporting does not weigh enough to prevent it from doing rising. The balloon will rise to the same equilibrium height.[br][br]What quantity is flowing? [i.e., what quantity corresponds to [i][color=#6aa84f]liters/min [/color][/i]or [color=#0000ff][i]dollars/year[/i][/color] ?[/b][/color]