Leigh and David are playing a game of laser tag. David has managed to climb aboard a mining cart travelling along a parabolic track, while Leigh has been caught exposed trying to walk along a narrow path on the edge of a cliff. As long as David is in the cart he can only fire his laser at right angles to the direction of motion. If Leigh is caught in David's sight line he is doomed, but he can see a gap between the cliff and some rocks ahead of him. If he can get far enough into the gap David will not be able to see him and tag him. A rough schematic of the situation is sketched below. David Cart Tracks Druction of Frel Rocks Leigh Cliff Edge Safety This situation can be modelled on the cartesian plane as follows, where all quantities are measured in SI units: . Let s = [0, ) be a variable representing time. Leigh's position is represented by a point that travels along the x-axis, starting at the origin, with a positive constant speed v. David's position is represented by a point that travels along the parabola y = x-6x + 14 so that at times the x-coordinate is s. The rocks are represented by a line segment x = 10 for y 3. (a) Determine the (x, y) coordinates for Leigh's and David's position at times. (b) Find the equation of a normal line to the parabola, in terms of s, which passes through David's position. (c) Find the point H where the normal line found in (b) intersects the x-axis in terms of s. (d) If there were no rocks available for Leigh to hide in, would he be able to escape from David without being tagged? Justify your answer. (e) Use DESMOS to draw a graph with all the features of this model. Your drawing needs to be able to represent variable values of time and include labels (not hand-written) for every feature. Include a screenshot of this graph with a viewing window x = [-2, 20] and y = [-2, 15] in your solution. (f) Calculate the minimum speed that Leigh needs to travel at in order to escape. Justify your answer. (Hint: Think about the value of s where David's line of fire hits the bottom of the rocks.)