Final answer:
Fixed-point iteration can solve for x in the given equation with p = 3 and k = 0.05, but convergence of the iteration functions is dependent on the initial guess and the nature of the function. Two possible iteration functions were provided along with an explanation on checking for convergence.
Step-by-step explanation:
Fixed-Point Iteration for Solving Equations
The equation k = x/(1 - x) √(2p / (2 x)), given p = 3 and k = 0.05, can be solved using fixed-point iteration. First, we need to rearrange the equation to bring x on one side, obtaining one or more iteration functions. For example, we might isolate x on the left side and get the iteration function g(x) = k(1 - x)/√(2p/(2 x)). Another iteration function could be g(x) = 1 - (√(2p/(2 kx))).
Convergence of these functions depends on the choice of the initial guess and the nature of the function itself. Some initial guesses might cause the process to diverge or oscillate without converging. In order to check convergence, one could plug an initial guess into the iteration function and repeatedly apply the function to see if the values approach a steady number.
The first iteration function may converge to a solution depending on the initial guess, but we must ensure that the guess is within the function's radius of convergence. Similarly, the second iteration function may converge too, but again, the initial guess plays a crucial role, and it is necessary to verify the range for which the function is valid and convergent.