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#