Ladder Homework, Juiy 20th, 2006

 

1. Tom sees a ladder against a wall (in an almost vertical position).  He pulls the base of the ladder away from the wall by a certain amount, and then again by the same amount, and then again by the same amount, and so forth.

 

  1. Simulate this situation with a 12-inch ruler against a wall or some other surface. Does the amount by which the top of the ladder drops down get bigger, get smaller, or does it stay the same? Set up a table of values and use these values to justify (without using the idea of rate-of-change) your response.
  2. Construct a graph of this situation using data points that you collected. Label three points on the graph and state what each point conveys about the ladder’s position.
  3.   Define a function that characterizes the position of the 12 foot ladder as the base of the ladder is being pulled away from the wall at a constant rate and the top of the ladder is falling to the floor.
  4. Graph the function using graphing calculator. Move the cross-hair on the graph so that the amount by which the bottom of the ladder was pulled away from the wall varies. As the distance of the bottom of the ladder from the wall varies from 0 to 5, what specifically can you say about how the distance the top of the ladder is from the floor varies? (Use precise covariational language in your response.)

 

 

2.         A tank is made from two hollow globes that are connected by a hollow stem, as shown in the figure to the right. Water is being poured into the globes from the top.

 

a.     A twisted, flat, rod will be placed into this bottle so that it stands at the bottom of the lower globe and ends at the top of the upper globe. A float, slotted in the middle, that cannot turn moves up the rod, so that the rod turns as the float moves upward. The rod must have a total twist of 360° as the water raises the float from the bottom to the top of the tank. Sketch a graph of “amount of turn” as a function of the water’s height so that the rod turns at a constant rate with respect to the volume of water in the tank (i.e., same amount of volume produces same amount of turn).

 

 

b.     Explain how you constructed your graph, using language that refers to the varying quantities in the situation. (Remember to speak meaningfully and use landmarks as appropriate.)

 

(We’ll save the problem that Lisa was curious about until next week.)