Two people, Kim and Caroline, live at different places along the same road. They decide to meet at a particular point on the road. They each agree to start out from home at the same time and agree on a meeting time.[br][br]Choose a meeting time and a meeting place by dragging the BLACK dot to your choice of point on the graph.[br][br]You can choose positions along the road for Kim and Caroline’s homes by dragging the RED and BLUE dots. You can vary their speeds using the RED and BLUE sliders.[br] [br]Write a function that describes how far each of them has traveled at time t after they started out. What does your function depend on?[br][br]Write a function that describes how far apart Kim and Caroline are at any time t after they start out. What does your function depend on?[br][br]Two other people, David and Roger, also live along the same road but at different places. They would like to arrive at the meeting place at the same time. How can they arrange to do that?[br][br]Is it possible for anyone who lives anywhere along the road to arrive at the meeting place at the same time?[br][br]The principal decides to send a school bus to collect those who have gathered at the meeting place at the meeting time. Check the “school bus” check box. You will now see two sliders that let you adjust the time the school bus leaves and the speed at which it travels.[br][br]Write a function that describes when the school bus must leave and how fast it must travel in order to get to the meeting place at just the right time. What does your function depend on?[br][br]Finally, here is a problem, taken from Polya, that you can explore in this environment. A heads from home to B's house, delivers a package and returns home immediately. At the same time that A leaves her house, B leaves her house heading toward A's house delivers a package and returns home immediately. They meet the first time [i][b]a[/b][/i] meters from A's house and for the second time at [i][b]b[/b][/i] meters from B's house. [br][br]1. How far apart are A's and B's houses?[br]2. If [i][b]a[/b][/i] is 300 meters and [i][b]b[/b][/i] is 400 meters, who walks faster?