Question-   LO3 Approximate solutions of contextualised examples with graphical and numerical methods Task 3.0 3.1 (a) The voltage across a capacitor at a time t is given by vc = 120(1 – e-t/CR), where R = 47kΩ and C = 15 uF. Sketch the graph of voltage ag…

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Solution-

LO3 Approximate solutions of contextualised examples with graphical and numerical methods Task 3.0 3.1 (a)The voltage across a capacitor at a time tis given by ye= 120(1 — eV), where R= 47kQand C = 15 uf. Sketch the graph of voltage against time and estimate the value of the voltage when t =2s after being connected toa dc supply. (b)Atmospheric pressure p was found to vary with altitude, /as shown in the table opposite Show that this approximately follows the relationship p = ae"and by plotting a suitablearaph, find the constants a and k. Altit Pressure ude 500 73.9 150 0 68.42 300 0 61.6 500 0 53.56, 800 0 43.41

3.2The solution to a differential equation associated with the path taken by a projectile for which the resistance to motion is proportional to the velocity is given by :- Sete *)+x-25 Use (i), the bisection method and (ji) the Newton's method to determine the value of x, correct to 2 decimal places for which the value of y is zero. 3.3An alternating current i has the following values at equal intervals of 5 ms charge.ain coulombs is given by 30x107* a=[ idt o

Deteremine the charge stored in coulombs using (i)the trapezoidal rule an (ii) Simpson's rule. time (ms) | Current(A) 0 0 5 48 10 941 15, 12.7 20 8.8 25, 35 30 0 3.4 (a) The mathematical model of a system is given by the differential equation ®@ = xa+9-p ——(a) Using analytical method the solution of the above differential equation is 2=3t+, ett. Obtain the numerical solution of the differential equation (1)in the range 1.0 (0.2) 2.0 given the initial conditions that at ¢=1, when p =4. When x=1.6, compare the accuracy,correct to 3 decimal places, of the Euler and Euler — Cauchy methods. (b)Critically evaluate the accuracy and the applicability of the above numerical methods.

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