Problem 8: [35 pts] (Donuts: The Final Frontier) We've climbed many mathematical mountains this semester, so here's a sweet final problem to wrap things up! The top half of the infamous "Mount Donut" is modeled by the surface z = f(x, y), where f (z, y) = 19 - (V22 + 32 - 4). A. [10 pts] Find the domain and range of f(x, y), and sketch the domain on the graph below. Make sure to simplify your final answers and show your work. 7-6 5-4 3-2-1 1 2 3 4 56 7 B. [5 pts] Calculate fx(x, y). If you do it carefully, you should find that x (Vx2 + yz - 4) fx(x, y) = - Vaz + yz. 9 - (Vx2 + yz - 4) ? Now, suppose that there is a donut shop at the point (4, 0, 3) on the surface. Jim starts at the point (7, 0, 0) and hikes up the surface above the x-direction in the ry-plane. A vector-valued function that traces his position in time is given by r (t) = (7 - 2t, 0, V12t - 412) , which is valid from t = 0 until the time Jim first reaches the donut shop. The rest of the parts of this problem refer to the curve parameterized by the vector-valued function above. C. [10 pts] Set up an integral that gives the distance Jim travels during his hike. Then, evaluate the integral by using technology and give the distance Jim travels to 1 decimal place. Make sure to state what you used to evaluate the integral. D. [10 pts] Show that Vf(5, 0) is orthogonal to the level curve of f(r, y) through (5, 0). Hint: A parameterization for this level curve should be part of your solution