5. Consider a royal's legacy L as a function of time t (in years) that royal spent as regent. It is expected that the longer one rules, that their legacy increases at a rate proportional to their current legacy (more famous for being famous). That is, we expect to model L(t) by L'(t) = mL(t) for some constant m. (Please turn over.) (a) We expect L(t) = Af(t) where A is constant and f(t) is some function in t. State mathematically what we expect f(t) to be. (Hint: This may vary depending on another base constant.) (b) Describe units of L as crown points. Queen Elizabeth II ruled from her coronation in 1952 to her death in 2022. Suppose that in 1952, her Majesty had 20 crown points, and that by 2022 had 5,000 crown points. Find function L(t) by solving for all constants. (c) Explain why this model for L(t) may be unrealistic.