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Two Terrains

Grade:
9-12
Standards:
Algebra
Math Content:
Algebra

Some vehicles fly down the highway, while others are dirt-track specialists. This activity allows you to explore the relationships between, distance, rate, and time. Can you identify the best path for a vehicle? Try to predict what will happen if you set on‑road or off‑road performance measures.

The Display 

For each car, the distance is displayed on the top, the raceway below that, the elapsed time to the left, and the speed at the bottom. Throughout on-road measures are displayed in red, and off-road measures are displayed in blue. For elapsed time, total elapsed time (on-road + off-road) is displayed in yellow.

Choose your mode at the bottom left of the tool. Manual Mode allows you to set distances and speeds. Random Mode restricts certain measures depending on the setting chosen.

Setting distance:

  • You can set ON-Road Distance by dragging the yellow dot left or right. The OFF-Road Distance is the distance from the yellow dot to the destination.
  • Alternatively, you can set either distance by entering values in the top boxes. ON-Road Distance must be less than or equal to 10 mi, and OFF-Road Distance must be greater than or equal to 10 mi.

Setting speed:

  • Set the ON-Road Speed and the OFF-Road Speed by moving the green slider below the respective speedometer.
  • Alternatively, you can set speed by entering values in the boxes. Speed must be a value between 0 and 120.

After adjusting your settings, click the GO button to start the race. The winning total elapsed time (in yellow) will flash.

Random Mode 

  • If you choose one of the three speed settings, then the speeds will be randomly chosen according to the choice you make. You can still change distances.
  • If you choose one of the two distance settings, then the distances will be randomly chosen according to the choice you make. You can still change speeds.
  • Before or after you choose the simulation, you can choose new random values by clicking the New button on the bottom, right.
 

Choose Random Mode, and select the ON‑Road < OFF‑Road Speed option. Change the ON‑Road Distance and OFF‑Road Distance for one of the cars so that your car beats the other car. Click GO. Did your car win? If your car didn't win, try changing the distances until it does. What is the best path for a vehicle that is faster off‑road?

Change to Manual Mode, and set the speeds to the values below. For each set of speeds, what are the best on-road and off-road distances?

   ON‑Road Speed      OFF‑Road Speed   
8040
6030
2x x 

In general, what are the best distances if the on‑road speed is twice the off‑road speed?

What is the ratio of on‑road speed to off‑road speed when the best path requires driving half as far on‑road as off‑road?