Question

How do you use the square roots of 144 and 169 to estimate the square root of 154?

Answer 1

Since `sqrt144 = 12` and `sqrt169 = 13`, `f(x) = x^2` is quite steep by this point, and `Deltax = 1` is a small change in x, it is a reasonable assumption to estimate this portion of the graph as linear, and say that:

Difference:
`169 - 144 = 25`

Percent of the way through the difference:
`154 - 144 = 10`
`=> 10/25 = 40%`

Thus a nice estimate for the square root of 154 is 0.4 + 12 = 12.4. The true square root is 12.410, with a -0.0780% error.