Estimate square root of 90 to the nearest hundredths place

The method for doing this will depend on the required accuracy of your estimate.

Using linear approximation between 9^2 = 81 and 10^2 = 100, the estimate is
.. 9 +(90-81)/(100-81) = 9 +9/19 ≈ 9.47

Using one round of "Babylonian Method" on the above estimate,
.. (9.47 +90/9.47)/2 ≈ 9.486 ≈ 9.49 . . . . . . actually accurate to the hundredths place

Using derivatives and linear approximation,
.. √90 ≈ √81 +1/(2√81)*(90-81) = 9.50
or
.. √90 ≈ √100 +1/(2√100)*(90 -100) = 9.50

Assume no calculators, no paper, no pen, all in the head.

90=9*10, so 9.5 is a good approximation.
in fact, 9.5=9*10+0.562=90.25
We will use this initial estimate x0=9.5 to improve to accurate to the hundredth (and more).

Newton's method gives a better estimate by the following formula (all done in the head)
x1=x0-(x0^2-90)/(2*x0)
=9.5-(90.25-90)/(2*9.5)
=9.5-0.25/19
=9.5-0.013125   [in the head, 0.25/19=0.25/20+5%=0.125+0.0625=0.013125]
=9.486875
=9.49  (to the hundredth)

Exact value = 9.48683298... 

The answer is 1532.75