(Originais Teachy 2023) - Question Medium of Mathematics

A technology company specialized in cloud computing is designing cones for the assembly of antennas for a new satellite data transmission service. Each antenna, in a cylindrical shape with a base at the top, must be surrounded by a cone to direct the transmission at a specific angle towards the atmosphere. The critical dimensions of the cone are designated to ensure that the transmission beam is ideally directed, avoiding signal losses. The cone's opening angle, called 'solid angle,' is a measure of the space covered by the transmission beam in the atmosphere, and is calculated in terms of its aperture and height. Knowing that the relationship between the base radius and the height of a right cone is essential for calculating the solid angle, and that a measurement error in these dimensions could result in significant service degradation, determine the metric relationship between the base radius and the cone height that ensures the accuracy of the solid angle for the assembly of data transmission antennas. Also, consider that signal loss is unacceptable and must be minimized, and that data transmission is affected by atmospheric refraction, which takes into account the Earth's curvature and the variation of air density with height.