The Problem:
A farmhand needs to take a bucket of water to her cow. She must go to the river first and fill up the bucket. What path should she take to minimize the distance to the cow, if she must stop by the river first?
You can use a Dynamic Geometry system to help you solve this problem, saving the farmhand time. First, you need to make a sketch to model the situation. Use a line segment to represent the river. Put a point where the cow might be and another point to indicate the location of the farmhand. If you are comfortable with clip art, you could find a picture of a cow online, copy and paste it into your drawing to make it more realistic. If you do this, make sure to use free, not copyright protected, clip art.
Now you need to make your first guess that is where on the river should the farmhand go to minimize the distance to the river and then to the cow? In order the sketch your guess, you will need locate a point, E, on the river. Place point E on the river, determining what you think would be the shortest path. Then, draw segments from the farmhand, to the river and from the river to the cow.
In order to test your guess, you need to know some measurements. Using the measuring menu, measure segments on your path and calculate their sum. Be sure the sum and measurements are displayed to an appropriate precision setting to make your decision clear.
Now, test your guess. Place the cursor on point E; hold down the button on the mouse and move point E along the segment until the sum is minimized. Answer the following questions:
1. Where do you think the farmhand should go the river, in order to minimize the distance he must travel? (Describe your solution on your sketch. Save your sketch and upload it to Geogebratube.org.)
2. How do you know this is the shortest path?
