Question 1

Use Newton Newton-Raphson method to determine numerically the solutions of the following equations: (you can assume of tolerance of 0.00001 is acceptable)

1. 4x4-2x²+2x-1 = 0

2. 3x³ 2x²+x-1 = 0 Questions 2 Consider a vessel with a volume Vo=0.5 m³ and uniform cross-sectional area (A = 0.25 m²). Initially(i.e. at t=0) there is 100L of fluid in the tank, fluid is introduced into the tank by opening a valve at time t= 0, and the flow rate is F₁. There is a tiny hole at the bottom of the tank which releases the fluid at a constant flow rate of F2 which has not been discovered. 1. Write done an equation at allows determination of the height of the fluid in the tank and at any given time t.2. Given that, F₁ = 0.005 m³/min and F2 = 0.0001 m³/min and the cross-sectional area is 0.25m². Use the Euler's method with a time step of 0.25 min to estimate the height of the fluid after 2.5 min. 2. Given that, F₁ = 0.005 m³/min and F₂ = 0.0001 m³/min and the cross-sectional area is 0.25m². Use the Euler's method with a time step of 0.25 min to estimate the height of the fluid after 2.5 min. Question 3 A product C is formed from A through the formation of an intermediate product B according to the reaction: A \longrightarrow B \longrightarrow C Where k1 = 0.08tanh(27)min-¹and k₂ = 0.01 min-1. At time t = 0 the concentration of A is 30 mols whilst concentrations of B and C are both 0 mols. • If the concentrations of A, B, and C at any given time are represented by X1, X2, and X 3 write down differential equations for determining the rate of change of concentration of A, B, and C.respectively,

• Using the Euler method, determine numerically the concentrations of A, B, and C after 25 minutes of the reaction using the time step of 2.5 minutes.