In any case, the volume in question will be $LW(d + H/2)$, so all you need to do is find that $y$-coordinate $d$. For example, lets pretend that your rectangular prism with a volume of 72cm has a length of 3cm and a width of 3cm. I suspect that this calculation can be avoided by some symmetry-argument. The volume of a rectangular prism is found by multiplying its length by its width by its height: V (L) (W) (H) If you know any 3 of these values, then you can easily solve for the unknown. The equation of the surface of the water has the form Let $d$ denote the $y$-coordinate at which the surface of the water hits the $y$-axis. We ultimately won't need it for the final answer, but if you're interested this $\mathbf n$ can be calculated from your Euler angles as the second column of the rotation matrix corresponding to your Euler angles. The formula for finding the volume of a rectangular prism is the following: Volume Length Height Width, or V L H W. Let $\mathbf n = (n_1,n_2,n_3)$ denote the normal vector to the surface water (relative to your intrinsic coordinate system of the prism). I assume that your origin is taken to be in the center of the prism. This formula also assumes that the entire "bottom" of the prism (relative to your chosen axis) is submerged. In this task, students will learn that one box of Girl Guide Cookies contains 24 cookies. Students will measure the volume of the prism using cookies as their non-standard unit of measure. V = (d_z + L/2)WH, \qquad V = L(d_x + W/2)H Students will determine the volume of 8 boxes of Girl Guide cookies organized in a 2 x 2 x 2 rectangular prism. In this exceptional case, you could use one of the other axes to get the formulas Note: This formula fails when the prism lies such that the $y$-axis is parallel to the surface of the water, in which case the $y$-axis intersection is non-unique. If $\theta$ is the angle that the $y$-axis of the prism makes with the vector pointing straight up, then $d_y = h/\cos \theta$. Answer: $V = LW(d_y + H/2)$, where $d_y$ is the $y$-coordinate at which the surface of the water hits the $y$-axis.
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