Question is in three parts and asks for a relation between two space time intervals, an explanation of length contraction and time dilation, then an application of proper time.
i) Two inertial frames O and O' are in standard configuration. Write down the two equations relating spacetime intervals Delta x' and Delta t' in O' to the corresponding intervals Delta x and Delta t in O. Then write the corresponding two inverse relations for Delta x and Delta t in terms of Delta x' and Delta t', explain any special symbols you use.
ii) Now consider these four relationships. Explain how the length contraction relationship follows immediately from one of them, and the time dilation relationship follows immediately from another. Include all essential steps in the argument, in particular stating the reasons for your choice of equation.
iii) The time dilation relationship just derived applies to uniform motion. Briefly explain how the concept of proper time permits the time dilation relationship to be applied to, the half-life of a muon circulating in a magnetic storage ring.