**Ladder Homework,
Juiy 20 ^{th}, 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.

- 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.
- 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
ladderÕs position.
- 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.
- 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 Òamount of turnÓ as a function of the waterÕs 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.)

(WeÕll save the problem that Lisa was curious about until next week.)