Final answer:
The measure of the angle at the base of a trapezoid drawn in a semi-circle that maximizes its area is 90 degrees because the base is the diameter, making the trapezoid effectively a right-angled triangle with its height reaching the semi-circle's arc.
Step-by-step explanation:
To find the measure of the angle at the base of a trapezoid drawn in a semi-circle that maximizes its area, we must consider the principles of geometry related to circles and triangles. Because the base of the trapezoid is the diameter of the semi-circle, every angle inscribed in the semi-circle and subtended by the diameter is a right angle (90 degrees) according to Thales' theorem. Therefore, the angle at the base of the trapezoid that maximizes the area is a right angle or 90 degrees.
This is similar to the concept where the optimum angle at which a projectile should be launched to cover maximum distance is 45 degrees. Moreover, in questions involving physics such as launch angles of projectiles, magnetic fields, inclined planes, and circular motion, specific angles result in maximum or zero values for certain properties or outcomes.
The trapezoid, in this scenario, effectively becomes a right-angled triangle, wherein the base is the diameter and the height reaches the semi-circle's arc. The maximum area of the triangle and thus the trapezoid is achieved when the height is maximized, which occurs at a right angle.