As you approach, your ship maps the mountain, Mt Honesty, and provides height as a function of the distance away from the centre of the mountain. Taking the origin to be level with the base of the mountain and centred inside the mountain the height, m, can be described as: m(x, y) = (x² + y²) exp (x²+ y²), 4 -10 ≤ x ≤10, -10 ≤ y ≤ 10, where x and y are the number of kilometres from the described origin. 1. Using the same coordinates, you land at (x,y) = (√3, √√3) facing a direction You remember from space training that your parallel to the vector u = landing ramp can only operate on a surface that has a gradient less than 10%. [Note: gradient here refers to the idea of percentage gradient, which calls an even rise over run as 100%. For our case, this just means whatever the slope is, multiply it by 100 to see what the percentage gradient is.] (a) By finding the slope in the direction that your ship is facing, determine whether you can safely use your landing ramp. (b) Once you disembark, you want to know which way to go to get to the top of the mountain, because that is where the mineral will be. You know that the maximum slope will tell you where is most up. Determine the maxi- mum slope, and in which direction it occurs to figure out which way to start trekking to reach the top of the mountain. 2. You have reached the top of the mountain and found that there seems to be a pool at the top. You can see at the bottom of the pool there is a glowing light. That must be the M2Z2B1 that you are looking for! You notice there is a current in the pool. You use a velocity probe and your computer maps the velocity field to tell you the field looks like: V = -2yi + 2xj, using the same coordinate system as before. In order to safely navigate to the mineral, you need to float a beacon on the surface of the pool. Given the velocity of the fluid, your beacon is going to be moving. You don't want the beacon to get stuck somewhere in the middle of the pool, and you want to know if it will be spinning around as it moves. (a) By determining the divergence of the velocity field, determine whether the beacon will be "sucked" in to anywhere in the pool, or whether it will just travel around a consistent path. (Remember, positive divergence values mean that there is a "push" away from that location and negative values mean there is a "pull" towards that location.) (b) By determining the curl of the field, determine whether your beacon will be spinning, and if it is which direction it will be spinning. (Remember that you can use the "right-hand rule" to determine the direction of rotation from the curl, positive is out of the page, negative is into the page.) 3. Having safely deployed your beacon you dive down, reach the mineral, and bring it back to the surface. Now you need to figure out how much you have found. Taking it back to your ship, you measure it and find that it appears to be shaped like a tetrahedron. By aligning it to some axes, it looks like you can define the volume of mineral by the region enclosed by the planes x = 0, y = 0, z = 0, and z = 1-x+y. Knowing that the density of the mineral is homogenous throughout the volume, we are just interested in the volume of mineral we have. Calculate this volume.