Quantitative Methods Answers

The bacteria concentration in a reservoir varies as

e^(t )-t^3/6-e^(0.3 t)-t^2/2 -t

where t is the time in seconds. Use the Newton-Raphson method to estimate the
time required for the bacteria concentration to reach 1 (correct up to 2 decimal
places)

The time versus velocity data of a particle is given in the table below. Use Lagrange’s
interpolation formula to find the distance moved by a particle and its acceleration at
the end of 3 seconds.
t: 0, 1, 2, 5
v: 2, 3, 12, 147

The time versus velocity data of a particle is given in the table below. Use Lagrange’s
interpolation formula to find the distance moved by a particle and its acceleration at
the end of 3 seconds.
t :0 ,1 ,2, 5
v: 2, 3 ,12 ,147

The current in an electric circuit is given by
i=t sin t
where t is the time in seconds. Using the bisection method, estimate the time
required for the current to reach 1 amp (correct up to 2 decimal places)

a)Use Newton's method to approximate the root of the equation xsinx =1 in [0,pi/2].Taking x=1 as the initial guess calculate x1,x2,x3(Work to at least 6 decimal places)
b)Show that the equation x^3-3x^2+2x=1 has no root in the interval [-1,1]

Q: The equation x^3+2x^2-5=0 has a positive real root in the interval (1, 2). Write a fixed point iteration method and show that it converges. Starting with initial approximation x=1.5 find the root of the equation. Perform two iterations.

A book binder has one printing press, one binding machine and manuscripts of seven
different books. The time required for performing printing and binding operations for
different printing and binding operations for different books are shown below:

Book 1 2 3 4 5 6 7

Printing Time 20 90 80 20 120 15 65
(Days)

Binding Time 25 60 75 30 90 35 50
(Days)

Find the optimum sequence of processing of the jobs that minimises the total time required.
Also compute the optimal time required.