plz solve all parts correctly

Problem 3: In this problem. we consider the simplified) simulation-based design optimization of the control mechanism for a prototype of a robotic lawn mower. You consider a (virtual) rectangular lawn 20 meters in the x-direction, and 30 meters in the direction. The lawn mower generally moves straight ahead with a spood of 1 meter per second (just to keep things simple), and only changes direction if either of the following happens: When hitting the boundaries of the lawn, the mower does an inelastic bounce. le. if the mover hits the left or right hand side boundaries, the sign of the velocity in the r-direction is switched, and if the mower hits the lower- or upper boundaries, the sign of the velocity in the y direction is switched. . At exponentially distributed times (since the last change in direction), the mower randomly chooses a new direction. These exponentially distributed times have expectation Etime. The effect of choosing different values of Etime is illustrated in Figures 1 and 2. The left panel in both figures show the centerpoint of the mower) 1800 seconds worth of mowing. In Figure 1, Etime = 1.1.e. random direction changes occur on average every second. In Figure 2. Etime 1000 which effectively turns of the random direction changes. In order to assess the relative merit of different values of Etime, the complete lawn is divided into 20 cm by 20 cm grid cells! Subsequently, it is counted how many of these cells were visited at least once by the mower. The right panels in Figures 1 and 2 illustrates this, where red-colored cells are visited. Finally the proportion of visited cells (ie. visited cells/total number of cells) is To reflect that the actual cutting device of the mower is around 20 cm wide 4 trace of town mowe grid colhe R 2 Figure 2: Realization of the lawn mower process with Etime = 1000.0. Only around 8% of the grid cells were visited taken as a performance measure In Figure 1, 37% of cells were visited, whereas only 8% of the cells were visited in Figure 2 R-code for simulation the lawn mower process is provided in the file lawn. R. The file consist of a single function with signature simulate.laun