Final answer:
To find the main diagonal of a box, determine the dimensions from the given volume and face areas, then apply the Pythagorean theorem in three dimensions to find the diagonal length.
Step-by-step explanation:
To find the length of the main diagonal of a 3D rectangular box with a volume of 10sqrt5, and two faces with areas 2sqrt5 and 5sqrt2, we need to use the properties of a rectangular box and the Pythagorean theorem. The dimensions of the box can be represented as length (l), width (w), and height (h). The volume of the box (V) is the product of these dimensions (V = lwh), and in this case, it is given as 10sqrt5.
Let's say the two given faces correspond to the areas lw (2sqrt5) and wh (5sqrt2), respectively. We can find the missing dimension by using the third face area, which must be lh. Since we have V, lw, and wh, we can find lh by dividing the volume by the width (V/w = lh). Once we have lh, we can determine the height h by dividing lh by l.
To find the length of the main diagonal (d), we use the Pythagorean theorem in three dimensions: d² = l² + w² + h². After finding the lengths l, w, and h from the given areas and volume, we plug them into this equation and solve for d to get the length of the main diagonal.