![]() We need to do this for each pair of a, b, and c, and then double the result to get all 6 faces. You can find the area of a parallelogram by taking the cross-product of two of its edges as vectors and then taking the length of the resulting vector. Now let's calculate the surface area of X. This product has to be equal to 355, but we notice that c_1 and c_2 don't enter into the volume formula at all. The cross product of a and b is (0, 0, a_1 * b_2), and then taking the dot product of this with (c_1, c_2, c_3) gives a_1 * b_2 * c_3. (In fact, this is true for any parallelepiped.) Let's use the vector formulation. Now, these three VECTORS a, b, and c can be used to find the volume of the rectangular prism X that they generate. Reflecting X if necessary, we can assume without loss of generality, that a_1, b_2, and c_3 are all strictly positive. (b_3 is zero because it lies in the x-y plane and b_1 is zero because the face is a rectangle.) Finally, the third edge coming from the origin will end at a general coordinate c = (c_1, c_2, c_3). Next, we'll have one of the adjacent rectangular faces be parallel to the x-y plane, and so its other edge must be of the form b = (0, b_2, 0). ![]() Then we'll line up one of the edges to run along the x-axis, and so have coordinates a = (a_1, 0, 0). We'll pick one of the vertices of X to be the origin (0,0,0). So now we suppose we have our rectangular prism X sitting in space. That would then mean that the cube minimizes surface area for a fixed volume among this class, and so if the cube can't get to a surface area that low, then no rectangular prism can. ![]() I'm going to show that there is a cube of the same volume with equal or smaller surface area. To see this, suppose we have ANY rectangular prism with a volume of 355. As stated, a rectangular prism (even an oblique one) with a volume of 355 cannot have a surface area below 6 * 355 2/3, which was the 300.817. I'm a couple of years removed from my last math class, so I might be missing something completely. Decreasing the length of one side, on the other hand, would decrease the volume, which is already given. But it would also increase the surface area to some number greater than 289. Increasing the length of one side of the cube would keep it as an RP, but make it no longer a cube. I took that and figured out the surface area for a cube with a volume of 355. If the volume of a cube is 355, then the length of one side is the cube root of 355 (7.080699). The lowest possible surface area I can figure out for an RP with a given volume of 355 is something slightly larger than 300.817789971606. Friend's kid had a problem and I'm stumped.ĭealing with a rectangular prism (RP), not a cube.Ĭan the surface area of this RP be less than 289.
0 Comments
Leave a Reply. |