Bobsledding involves hurtling down a steep, windy, narrow track of ice in a sled, at higher speeds than allowed on most highways. For a track to be both challenging and safe, it requires extremely careful design. This assignment will explore some calculations relevant to the preliminary design of a track. Bobsled tracks are typically around 1.5 km long with numerous bends (as shown in the image below of the Sanki track). The steepest grade is around 15 to 20%. Paths for the track are chosen so that the natural slopes are used as much as possible. Part A: Track slopes A track is planned for a ridge whose topography can be approximated by h($,y) = 1 + 0.13-332+2$v-392 Where h is the height above sea level in kilometres,l :r is the distance east of the peak in kilometres and y is the distance north of the peak in kilometres. You may nd it useful to visualise this using Matlab with the following commands2 .0.05:1]; y = [-1:0.05:1]; [xx,yy] = meshgrid(x,y); h = 1 + 0.1*exp(-3*xx.\"2 + 2*xx.*yy - 3*yy.'2); mesthx,yy,h) to get a surface plot, or replace the last line by contour(xx,yy,h) axis equal to get a contour plot. 1. Find the gradient of h(:r,y). [2 marks] 2. Hence nd the magnitude of the steepest slope S (:13, y) at each point (:13, y) [1 mark] 3. Briey describe how you would nd the location of the overall steepest slope. (You do not need to do any calculations.) [2 mark] 4. Possible locations for the overall steepest slope are (0,0), (1/4, 1/4), (1/4, 1/4), (1/\\/, 1AA?) and (1/\\/, 1/\\/). Using the contour plot of Mr, 3;), explain which of these points has/have the steepest slope. [1 mark] A proposed section of the track is given in parametric form by 93(6) = 0.1 sin(107r) 5 and y(6) = 0.1 sin(1011') + 5, with 2(5) = h(m(),y()) and parameter values 0 S 5 S l. 5. Using Matlab or other software, plot this curve in the (:r, 3;) plane (in other words, plot the path of the track viewed directly from above). [1 mark] Find an expression for the tangent vector to this proposed track in the (I, y) plane (in other words only considering the :1: and 3; components). [2 marks] Use this tangent vector to nd the slope along the track 3(6). [3 marks] Plot the height of the track 2(5) and slope of the track 3(5) that you have calculated against 5, the horizontal distance of the centreline of the path from the peak. Be sure to include axis labels and units on your plot. [3 marks] If the steepest allowed slope is *02, are there any points where the slope is too steep on this track? [1 mark]