Final answer:
Using trial and improvement, it's determined that the approximate solution for the square root of 129 lies between 11 and 11.5. By repeated trials, one can conclude that 11.35 is a suitable approximation.
Step-by-step explanation:
To find the approximate solution to x = square root of 129 using trial and improvement, we start by choosing numbers to square that we think might be close to 129. We can start with 11, because 112=121, which is close to 129. If 11 is too small, we can try a slightly higher number, like 11.5. Squaring 11.5 gives us 132.25, which is just over our target of 129. So we know our answer is between 11 and 11.5.
We can then try 11.1, where 11.12=123.21, which is closer but still under 129. Next, we might try 11.3 where 11.32=127.69, again closer but still not 129. Finally, trying 11.4 gives 11.42=129.96, which is very close to 129. The number 11.4 is slightly above our target, so we know that the square root of 129 is slightly less than 11.4. Through further refinement, we might decide that 11.35 is an adequate approximation for practical purposes, as 11.352\u2248 128.82, which is very close to 129.