square root of 10800 with formula
Answers
Answer:
sqrt(10800)' is 103.92304845413264.
To find the square root of S, do the following:
Make an initial guess. Guess any positive number x0.
Improve the guess. Apply the formula x1 = (x0 + S / x0) / 2. The number x1 is a better approximation to sqrt(S).
Iterate until convergence. Apply the formula xn+1 = (xn + S / xn) / 2 until the process converges.
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Answer:
Step 1:
Divide the number (10800) by 2 to get the first guess for the square root .
First guess = 10800/2 = 5400.
Step 2:
Divide 10800 by the previous result. d = 10800/5400 = 2.
Average this value (d) with that of step 1: (2 + 5400)/2 = 2701 (new guess).
Error = new guess - previous value = 5400 - 2701 = 2699.
2699 > 0.001. As error > accuracy, we repeat this step again.
Step 3:
Divide 10800 by the previous result. d = 10800/2701 = 3.998519067.
Average this value (d) with that of step 2: (3.998519067 + 2701)/2 = 1352.4992595335 (new guess).
Error = new guess - previous value = 2701 - 1352.4992595335 = 1348.5007404665.
1348.5007404665 > 0.001. As error > accuracy, we repeat this step again.
Step 4:
Divide 10800 by the previous result. d = 10800/1352.4992595335 = 7.9852169411.
Average this value (d) with that of step 3: (7.9852169411 + 1352.4992595335)/2 = 680.2422382373 (new guess).
Error = new guess - previous value = 1352.4992595335 - 680.2422382373 = 672.2570212962.
672.2570212962 > 0.001. As error > accuracy, we repeat this step again.
Step 5:
Divide 10800 by the previous result. d = 10800/680.2422382373 = 15.8766971425.
Average this value (d) with that of step 4: (15.8766971425 + 680.2422382373)/2 = 348.0594676899 (new guess).
Error = new guess - previous value = 680.2422382373 - 348.0594676899 = 332.1827705474.
332.1827705474 > 0.001. As error > accuracy, we repeat this step again.
Step 6:
Divide 10800 by the previous result. d = 10800/348.0594676899 = 31.029180363.
Average this value (d) with that of step 5: (31.029180363 + 348.0594676899)/2 = 189.5443240265 (new guess).
Error = new guess - previous value = 348.0594676899 - 189.5443240265 = 158.5151436634.
158.5151436634 > 0.001. As error > accuracy, we repeat this step again.
Step 7:
Divide 10800 by the previous result. d = 10800/189.5443240265 = 56.9787571085.
Average this value (d) with that of step 6: (56.9787571085 + 189.5443240265)/2 = 123.2615405675 (new guess).
Error = new guess - previous value = 189.5443240265 - 123.2615405675 = 66.282783459.
66.282783459 > 0.001. As error > accuracy, we repeat this step again.
Step 8:
Divide 10800 by the previous result. d = 10800/123.2615405675 = 87.6185706448.
Average this value (d) with that of step 7: (87.6185706448 + 123.2615405675)/2 = 105.4400556062 (new guess).
Error = new guess - previous value = 123.2615405675 - 105.4400556062 = 17.8214849613.
17.8214849613 > 0.001. As error > accuracy, we repeat this step again.
Step 9:
Divide 10800 by the previous result. d = 10800/105.4400556062 = 102.4278670749.
Average this value (d) with that of step 8: (102.4278670749 + 105.4400556062)/2 = 103.9339613406 (new guess).
Error = new guess - previous value = 105.4400556062 - 103.9339613406 = 1.5060942656.
1.5060942656 > 0.001. As error > accuracy, we repeat this step again.
Step 10:
Divide 10800 by the previous result. d = 10800/103.9339613406 = 103.9121367135.
Average this value (d) with that of step 9: (103.9121367135 + 103.9339613406)/2 = 103.9230490271 (new guess).
Error = new guess - previous value = 103.9339613406 - 103.9230490271 = 0.0109123135.
0.0109123135 > 0.001. As error > accuracy, we repeat this step again.
Step 11:
Divide 10800 by the previous result. d = 10800/103.9230490271 = 103.9230478812.
Average this value (d) with that of step 10: (103.9230478812 + 103.9230490271)/2 = 103.9230484542 (new guess).
Error = new guess - previous value = 103.9230490271 - 103.9230484542 = 5.729e-7.
5.729e-7 <= 0.001. As error <= accuracy, we stop the iterations and use 103.9230484542 as the square root.
So, we can say that the square root of 10800 is 103.923048 with an error smaller than 0.001 (in fact the error is 5.729e-7). this means that the first 6 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(10800)' is 103.92304845413264.