Consider the following expression. 8.30 x 10-5=x(0.100+42 We can solve for x usi

Question

Consider the following expression. 8.30 x 10-5=x(0.100+42 We can solve for x using a technique called successive approximations Step 1: If we assume that x is very small compared to 0.100 (such that 0.100 x 0.100) then our first approximation of x (let's call it x) can be calculated as 8.30 × 10-5-4 (0.100) 2 Number Express all answers to three or more significant figures t,E Step 2: Now, take your first approximation of x and plug it into the full equation 8.30× 10.5=x,(0.100+%)2 Number Step 3: Each successive approximation uses the value from the previous approximation. V Hint

Explanation / Answer

We are given the expression,

8.30 * 10-5 = x (0.100 + x )2

As per the approximatio we calculate x1

8.30 * 10-5 = x1 (0.100 )2   

x1 = 8.3 * 10-3 (3 significant figures)

Using x1 we calculate x2

8.30 * 10-5 = x2 (0.100 + x1 )2  

8.30 * 10-5 = x2 (0.100 + 8.3 * 10-3 )2

x2 = 7.08 * 10-3                         (3 significant figures)

Using x2 we calculate x3

8.30 * 10-5 = x3 (0.100 + x2 )2  

8.30 * 10-5 = x3 (0.100 + 7.08 * 10-3 )2

x3 = 7.24 * 10-3                                  (3 significant figures)

Using x3 we calculate x4

8.30 * 10-5 = x4 (0.100 + x3 )2  

8.30 * 10-5 = x4 (0.100 + 7.24 * 10-3 )2

x4 = 7.22 * 10-3                                 (3 significant figures)

Using x4 we calculate x5

8.30 * 10-5 = x5 (0.100 + x4 )2  

8.30 * 10-5 = x5 (0.100 + 7.22 * 10-3 )2

x5 = 7.22 * 10-3                                  (3 significant figures)

Thus, x4 and x5 gives the desired level of precision.

From above calculation we observe that x3 and x4 (7.2 *10-3) agrees to the two significant values.

Also, x4 and x5 (7.22 *10-5) agrees to three significant values.

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