Space Exploration: Lunar Rovers

Dorian and Elijah have been doing some work in preparation for going to the moon. Elijah is designing a lunar rover and Dorian is tasked with determining the best route in terms of time and distance for the rover to take on the moon. Here's the task:

(This task is from nasa.gov.)

By doing many calculations involving the Pythagorean Theorem and rates and graphing, he decided the best location was a little bit more than 11 km north of point P on the map. Along the way he derived the formulas below. In both equations, x represents the distance north of point P. The first equation calculates the total path length from Shackleton to de Gerlache to point B. The second equation calculates the amount of time it would take.  

After doing all that work, he plugged those equations into a graphing calculator to check his work. This is what the graph looked like. The red line is the path distance and the blue line is the total time. The x-axis represents the distance north of point P on the de Gerlache crater. This graph confirmed that the best location to collect rocks would be between 10.6 km and 12.6 km north of point P.

Elijah is currently at work designing the lunar rover. He's been working on a prototype which he is then scaling up to create a plan for building a life-size rover. His first prototype is complete and he is now working on a second prototype.  This is his first prototype along with his technical drawing. 

Here's his next prototype which is still under construction.