Day 2, July 18 Modeling Problems Homework

Homework:

1. The following problem is a typical homework problem from a precalculus book.

Similar to our classroom explorations of the box and circle problems I would like you to first identify and explore varying relationships; then determine what varying relationships should be modeled to respond to this question.

Problem Statement: You have 40 (linear) feet of fencing with which to enclose a rectangular space for a garden. Find the largest area that can be enclosed with this much fencing. Find the dimensions of the corresponding garden.

- What quantities vary in this situation?
- Identify two pairs of quantities that vary in tandem. Use the finger tool to imagine how they vary together. Provide a brief description of what you imagined and observed by using the finger tool for each pair.
- Using graphing calculator, construct a function that models this situation and create the graph in graphing calculator. Past your graphing calculator image into your solution.
- Describe
how the area changes as the length of one side of the garden changes from
*x*= 0 to*x*= 7. - Determine the dimensions of the garden when the area is 45 feet.
- What is the range of possible values that are possible for the width and length of the garden? Explain your answer in the context of the situation.

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. Sketch a graph that shows the height of water in this configuration as a function of the waterÕs volume in it.

b. A twisted, flat, rod will be placed into the configuration 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 volume so that the rod turns at a constant rate with respect to the volume of water in the tank.