| I finally came up with a decent group problem to use in my classes. The problem is printed below.
My students really enjoyed it and did a great job solving it. I appreciate the suggestions I received
and would love to hear any others. Thanks Marc
Recall the definition of a hyperbola. If a sound is recorded at three different locations, the location of that sound can be accurately found. If the difference in times is known, the difference in the distances is known, since sound travels at 1100 feet per second. If the difference in distances is known, then, according to the definition of a hyperbola, the point is located on one branch of a hyperbola. Three ships are in the Gulf of Obnoxia during an armed conflict. Ship A is the lead ship, so we will give it coordinates (0,0). Ship B is 4 miles directly to the east and Ship C is 3 miles directly to the north of Ship A. An explosion occurs and is recorded accurately on each ship. The explosion is recorded on Ship A 6 seconds after it is recorded on Ship B. The explosion is recorded on Ship A 4.8 seconds after it is recorded on Ship C. A. Draw a good size diagram of the coordinate system, labeling the 3 ships. Label all points in feet. B. Points A and B are the foci of a hyperbola. 1. Find the center of this hyperbola, using feet. 2. Using the difference in times, compute the difference in distances between the explosion and point A and the explosion and point B, in feet. 3. Use the information in #2 to find "a" for the hyperbola. 4. Use "a" and "c" to find "b" for this hyperbola. 5. Write the equation of this hyperbola in standard form. 6. The parametric form for the equations of a hyperbola with center at (h,k) are and horizontal
major axis are: C. Points A and C are the foci of a hyperbola. 1. Find the center of this hyperbola, using feet. 2. Using the difference in times, compute the difference in distances between the explosion and point A and the explosion and point C, in feet. 3. Use the information in #2 to find "a" for the hyperbola. 4. Use "a" and "c" to find "b" for this hyperbola. 5. Write the equation of this hyperbola in standard form. 6. The parametric form for the equations of a hyperbola with center at (h,k) are and vertical major
axis are: D. Using a graphical calculator. graph both hyperbolas above, in parametric form. Start the range set at: -250,250,5,-10000,20000,5000,-15000,18000,5000. Find the exact coordinates of the explosion based on the fact that it is a point located on both hyperbolas. Assume that the explosion occurs at the point that is farthest from point A. Give the coordinates to two decimal places. E. Find the direction and distance from Ship A to the explosion. |