3. (Inspired by Armendariz & Daniels, p. 40) In the early 1600s, Johannes Kepler proposed three laws of planetary motion, based upon data carefully collected by his mentor, Tycho Brahe. Kepler's 3'd law of planetary motion is sometimes called the Law of Harmonies. It states: desmos. com The square of the orbital period (P) of each planet is proportional to the cube of its mean distance from the sun (a), i.e. P2 = k * a3 where k is a constant which has the same value for all the planets in the solar system. a) Use a graphing utility to sketch two different graphical representations of the relationship between P and a, Online one with P on the horizontal axis, and (X- axis ) ii. one with a on the horizontal axis. (X - allis ) iii. Your sketches should show all interesting characteristics of the graphs. b) If you input a specific value for P into the equation p2 = k * a3, is there a unique value for a that satisfies the equation? Explain your response using both graphs. Page 2 of 3 of a set S into a set T consists of the set S, called the domain of the map, the ain, and a subset a of S x T (the graph) having the following two properties: there exists a t E T such that (s, t) Ea. , t') E a then t = t' Igebra [Abstract Algebra text]; Jacobson, 1985, p. 5. Page 2 of 6esc tab V Alvarez, J.A.M., Jorgensen, T., & Rhoads, K. (2018) Enhancing Explorations in Functions for Preservice Secondary Mathematics Teachers Project, The University of Texas at Arlington. c) If you input a value for a into the equation P2 = k * a3, is there a unique value for P that satisfies the equation? Explain your response using both graphs. d) Based on the context that this equation P2 = k * a3 describes, qualitatively what interval of values make sense for P and a? e) How does P2 change as a changes? a = 1 - a = 2 f) How does P2 change as a3 changes? a 3 = 1 - 2 3 = 2 4. (Adapted from Epperson, Jones, Jorgensen, Meeks, Olson, & Shahan (2004)) LORAN (Long Range Navigation) is a hyperbolic radio navigation system used to help locate ships or planes. The LORAN system uses three broadcasting stations to determine the location of ships at sea. The stations send out a repeated pulse, pulsing at specific time intervals. The navigator of the ship receives the pulses and determines the times taken to receive each signal. Based on those times, the distances from the three stations to the ship can be determined. Three LORAN broadcasting stations are located at the points S1, M, and $2. S1 is located 200 miles east of M. Sz is located 300 miles south of M. The navigator receives the pulse and finds the following distances: Point S