Final answer:
The exact length of the longest side of the wedge, which is in the shape of a right triangle with one 30° angle, is 2a. This is determined by the side ratio in a 30-60-90 triangle.
Step-by-step explanation:
The question relates to the application of trigonometry and the Pythagorean theorem in finding the length of the hypotenuse in a right triangle. When a right triangle has one angle of 30°, and one leg with length 'a', which is the height of the triangle opposite the 30° angle, the length of the hypotenuse can be determined using the properties of a 30-60-90 triangle.
In a 30-60-90 triangle, the ratio of the lengths of the sides opposite these angles is 1 : √3 : 2. Therefore, if the length of the side opposite the 30° angle is a, then the length of the hypotenuse is twice this length, or 2a. We don't need the actual measurements or the Pythagorean theorem here, as the predefined ratios of the sides in this special type of triangle give us a direct answer to the question.
The exact length of the longest side of the wedge, which is the hypotenuse of the right triangle, is therefore 2a.