How do you order the following from least to greatest without a calculator $sqrt102, 10, 3pi, sqrt99, 1.1times10^1, 9.099$ ?

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Answer 1 · 2017-04-23T08:28:58

$9.099, 3pi, sqrt(99), 10, sqrt(102), 1.1*10^1$

If $a$ is an approximation to $sqrt(n)$ then:

$sqrt(n) = a+b/(2a+b/(2a+b/(2a+b/(2a+...))))$

where $b = n-a^2$

If $a$ is a good approximation to $sqrt(n)$ then a better one will be:

$sqrt(n) ~~ a+b/(2a) = a+(n-a^2)/(2a)$

$102$ and $99$ are both close to $100 = 10^2$ , so using $a=10$ we find:

$sqrt(102) ~~ 10+2/20 = 10.1$

$sqrt(99) ~~ 10-1/20 = 9.95$

Note also that:

$3pi ~~ 3*3.14 = 9.42$

$1.1 * 10^1 = 11$

So the correct order of the given numbers is:

$9.099, 3pi, sqrt(99), 10, sqrt(102), 1.1*10^1$

How do you order the following from least to greatest without a calculator sqrt102, 10, 3pi, sqrt99, 1.1times10^1, 9.099sqrt102, 10, 3pi, sqrt99, 1.1times10^1, 9.099?