The cylinder has radius 2 m and length 10 m, whose axis is the vertical line through (2,0,0): we will put the bottom of the cylinder at z=0 and the top at z=10.

a) Suppose that the fluid level in the tank is 7'm on the left edge of the tank (where x=0) and i'm on the right edge (where x=4). Find the equation of the plane of the liquid, and use a double integral to find the volume of liquid in the tank. [Hint: you should use a "dy x iterated integral, where the bounds on y depend on x, and are given by the equation of the base of the cylinder.] Also find the angle at which the tank is tipped from its upright position. For consistency, assume that the tank in the Introduction above is tipped at a 0= angle. [Hint: use a "side elevation" sketch, and use trigonometry to find the angle between your plane and the plane z-5.] b) Suppose now that the fluid level is 7 m at the left edge of the tank, but that there isn't enough fluid to reach the right edge; instead, the fluid covers the base of the tank only out to the plane x=3. [You can produce a graph with the commands below: you'll have to paste them into another cell.] plane4 = ParametricPlot3D [ {X, Y , 7 -7x/ 3), {x, 0, 3), {y, -2, 2)] Show [cyl , plane4, AxesLabel + {X, Y , z) , ViewPoint -> {0, -3, 0)] Find the equation of the plane of the liquid and use a double integral to find the volume of liquid in the tank. Also find the angle at which the tank is tipped from its upright position. [Warning: the region of integration has changed -- if you reuse the bounds from problem 1, you'll include the "negative volume" from the region where z=0.] c) Suppose that the tank is tipped so far that the liquid touches both ends of the tank, touching the top (at z=10) out to the plane x=1, and covering the bottom out to the plane x=3. Find the volume of liquid in the tank using two double integrals. Explain why you can't just use one double integral. Lastly, find the angle at which the tank is tipped from its upright position. d) Suppose, finally, that the liquid touches the top of the tank out to the plane x=3, and reaches a level of 3 m on the right edge of the tank. Find the volume of liquid in the tank using two double integrals, and find the angle at which the tank is tipped from its upright position. ) Suppose that our tank is tilted at an angle of 80- from its upright position. Where could you mark the surface of the tank to indicate when the tank is exactly one-third full? [Hint: use trigonometry to decide which of the first four exercises most resembles this prob- lem. Calculate the volume in terms of some unknown variable, and use Mathematica's FindRoot command, which you can look up in the help browser.]