The temperature in a bar of length satisfies the heat equation du - 2004 where u(x, t) is the temperature in °c. the ends of the bar are maintained at a temperature of 0°c. hence u(0,t) = u(7.t) = 0 for all t. a) assuming a solution of the form u(r, t) = f(.r)g(t) show that = k where k is a constant. b) you may assume that only k < 0 yields non-trivial solutions and set k= -(p2) for some p > 0. applying the initial conditions show that p = n, n = 1,2, and that possible solutions for f(x) are = sin(nx) = 1, c) find all possible solutions g(6) for g(6)

The table below represents the displacement of a fish from its reef as a function of time: time (hours) x displacement from reef (feet) y 0 4 1 64 2 124 3 184 4 244 part a: what is the y-intercept of the function, and what does this tell you about the fish? (4 points) part b: calculate the average rate of change of the function represented by the table between x = 1 to x = 3 hours, and tell what the average rate represents. (4 points) part c: what would be the domain of the function if the fish continued to swim at this rate until it traveled 724 feet from the reef? (2 points)