Let R(A, B) be a relation with r > 0 tuples, and let S(B, C) be a relation with s > 0 tuples. We assume that A, B, and C have the same data type. For each of the following Relational Algebra expressions, in terms of r and s, determine the minimum and maximum number of tuples that the result relation can have. In other words, we are interested in the number of tuples the following Relational Algebra expressions can have at least and at most. The numbers have to be given by using the two variables r and s. Please note that you have to give precise and formal explanations for your answers.

a. [4 points] R u rho rum(S)
b. [3 points] (R Da R) pA R
c. [4 points] π4.dRx S)
d. [5 points] ơedR bd S)

The school track has eight lanes. each lane is 1.25 meters wide. the arc at each end of the track is 180. the distance of the home straight and the radii for the arcs in the 1st 4 lanes are given. s=85m r1=36.5m r2=37.75m r3=39m r4=40.25m part one: find the radii of lanes 5 through 8 of the track. show your work. part two: if max ran around lane one, how far did he run? show your work and explain your solution. part three: max wants to run a total of three laps around the track, choose two additional lanes (2-8) for him to run and find the distance around those two lanes. show your work and round to the hundredths. part 4: based on your lane choices in part three, what was the total distance max ran in the three laps around the track?