Math Problem Solver

In a data center, a server rack operates under a resonance condition that can interfere with the functioning of hard drives due to vibration-induced disturbances. The resonance frequency f in Hertz (Hz) depends on the rack's stiffness and mass, as well as the frequency of network and data processing loads. The resonance condition can be modeled by the equation:

R = (2k/m)-(f²(1+cos(f.T)))

where k = 3500N / m (stiffness of the rack), m = 200kg (effective mass of the rack with equipment), T = 0.8 seconds (average time interval of network and processing load cycles). To minimize disruptions, find the minimum frequency f (in Hz) that satisfies the resonance condition, ensuring that R >= 0.2

a. Use Bisection Method to solve for f within the range f in [1, 10] Hz, accurate to within 10-3.

b. Use Newton-Raphson Method to solve for f with an initial guess of f = 5 Hz, accurate to within 10-5.

c. Use Secant Method to solve for f with initial guesses of f = 1 Hz and f = 2Hz accurate to within 10-4.

d. Use Regula Falsi Method to solve for f with initial guesses of f = 1 Hz and f = 2 Hz, accurate to within 10-4.

e. After solving, discuss which numerical method (Bisection, Newton-Raphson, or Secant) performed better for this problem. Compare the accuracy, speed of convergence, and reliability of each method. (Consider aspects such as: 'How each method approached the root of the equation'; 'Sensitivity to initial guesses'; 'Suitability of each method for this type of problem'.)