Figure 2. Floating ball problem

The equation that gives the depth x to which the ball is submerged under water is given by

Use the Newton-Raphson method of finding roots of equations to find

a) the depth x to which the ball is submerged under water. Conduct three iterations to estimate the root of the above equation.

b) find the absolute relative approximate error at the end of each iteration, and

c) the number of significant digits at least correct at the end of each iteration.

Solution:

Let us assume the initial guess of the root of is

Iteration #1

The estimate of the root is

The absolute relative approximate error, at the end of Iteration #1 is

= 19.89%

The number of significant digits at least correct is 0, as you need a absolute relative approximate error of less than 5% for one significant digit to be correct in your result.

Iteration #2

The estimate of the root is

The absolute relative approximate error, at the end of Iteration #2 is

The number of significant digits at least correct is 2.

Iteration #3

The estimate of the root is

The absolute relative approximate error, at the end of Iteration #3 is

The number of significant digits at least correct is 4, as only 4 significant digits are carried through in all the calculations.

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