When intersected with the xy-plane, the tank will be an ellipse that goes through the points (0,0,0),

Fantastic news! We've Found the answer you've been seeking!

Question:

When intersected with the xy-plane, the tank will be an ellipse that goes through the points (0,0,0), (4,2,0), (4,-2,0), and (8,0,0). This ellipse is centered at (4,0,0), and every point on the ellipse satisfies the equation (1/4)(x-4)^2 + y^2 = 4. We will assume that the tank has a height of 10. The new tank looks like
܀ ܀ ܘ 1 ܀ ܀ v

a) Find a parametrization of the tank described in the preceding paragraph. Be sure to include your bounds on the parameters.

b) Suppose that the fluid level in the tank is 7 m on the left edge of the tank (where x=0) and 5 m on the right edge (where x=8). Find the equation of the plane of the liquid, and use a double integral to find the volume of liquid in the tank.

c) 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=4. 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.

d) 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=2, and covering the bottom out to the plane x=4. 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.

e) Suppose, finally, that the liquid touches the top of the tank out to the plane x=6, and reaches a level of 4 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.