rmine the angle a for which x =, derive the equation that must be solved and then determine the root usingollowing methods:3rUse the bisection method. Start with a= 0.5 and b = 1.5, and carryout the first four iterations.Use the secant method. Start with the two points a₁ = 0.5 anda₂ 1.5, and carry out the first four iterations.3.11 Problems3.4 The lateral surface area, S, of a cone is given by:S = πr√√r²+h²where r is the radius of the base and h is the height. Determine the radius of acone that has a surface area of 1800 m² and a height of 25 m. Solve by usingthe fixed-point iteration method with r = S/(√√²+h²) as the iteration func-tion. Start with r = 17 m and calculate the first four iterations.3.5 Determine the fourth root of 200 by finding the numerical solution ofthe equation x4-200= 0. Use Newton's method. Start at x = 8 and carryout the first five iterations.3.6 Determine the positive root of the polynomial x³ +0.6x² +5.6-4.8.(a) Plot the polynomial and choose a point near the root for the first estimate of the solution. Using New-ton's method, determine the approximate solution in the first four iterations.(b) From the plot in part (a), choose two points near the root to start the solution process with the secantmethod. Determine the approximate solution in the first four iterations.3.7 The equation 1.2x³ + 2x²-20x-10 = 0 has a root between x = -4 and x = -5. Use these valuesfor the initial two points and calculate the next four estimates for the solution using the secant method.3.8 Find the root of the equation √√x+x²= 7 using Newton's method. Start at x = 7 and carry out thefirst five iterations.α3.9 The equation x³ − x −e* − 2 = 0 has a root between x = 2 and x = 3.(a) Write four different iteration functions for solving the equation using the fixed-point iteration method.(b) Determine which g(x) from part (a) could be used according to the condition in Eq. (3.30).(c) Carry out the first five iterations using the g(x) determined in part (b), starting with x = 23.10 The equation f(x) = x² - 5x¹/3 + 1 = 0 has a root between x = 2 and x = 2.5. To find the root byusing the fixed-point iteration method, the equation has to be written in the form x = g(x). Derive two pos-sible forms for g(x) - one by solving for x from the first term of the equation, and the next by solving forx from the second term of the equation.(a) Determine which form should be used according to the condition in Eq. (3.30).(b) Carry out the first five iterations using both forms of g(x) to confirm your determination in part (a).893.11 The equation f(x) = 2x³-4x²-4x-20 = 0 has a root between x = 3 and x = 4. Find the root byusing the fixed-point iteration method. Determine the appropriate form of g(x) according to Eq. (3.30).Start the iterations with x = 2.5 and carry out the first five iterations.Solve the following system of nonlinear equations:Use Newton's method. Start at x =3.12 Determine the positive root of the equation cosx-0.8x² = 0 by using the fixed-point iterationmethod. Carry out the first five iterations.-2x³ + 3y² +42 = 05x² + 3y³69 = 0Chapter 3 Solving Nonlinear Equations1 and carry out the first five iterations.5x260) 1/3

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