On an aerial photograph, the surface of a pond appears as

spence11 , these problems can be confusing. We have a "scale factor" issue, as you note.

In one sense, however, when scaling up or down, linear change is one kind of step (ONE change in one length), and area change is a different kind of step (TWO changes, one for each length).

"...converting finding the area to in square inches and then converting from inches to miles [does not] work" because you are not converting from inches to miles.

You need to convert square inches (in the area you found for the pond on the photo) to square miles.

The scale factor here is
(2 miles: 1 inch) = 2

When converting an area, you have to use the scale factor, \(k\), twice.

Area conversion requires \(k^2\) because
-- the area of any two dimensional figure, circles included, is (length times length)

To convert area using the change from one linear unit to another (distance in inches -> distance in miles) ...
what you do to one length (scale it up), you must do to the other.

And to calculate area, we use two lengths. You converted only one of the two lengths.

You chose the correct scale factor of 2. But you need to multiply the area you found by \(k^2=2^2=4\).
You are short one factor of \(k\). Close!

(\(\frac{77}{128}\) * 4) = \(\frac{77}{32}\) = 2.4 (sq mi - see below)

Here is one way to look at it:

\(\frac{1in}{2mi}\) -- those are one-dimensional lengths. Lines.

Convert them to two-dimensional areas by squaring the number and the unit:

(1 in)\(^2\) = (1 in)*(1 in) = 1 sq. inch, or 1 in\(^2\)

(2 mi)\(^2\) = (2 mi)(2 mi) = 4 square miles, or 4mi\(^2\)

Below, area in square inches is converted to area in square miles.
Square inches cancel, leaving square miles as the desired result, but only because I multiply the original area by the scale factor twice:

\(\frac{77}{128}(in*in)\) * \(\frac{2mi}{1in}\) * \(\frac{2mi}{1in}\) = \(\frac{77}{32}\)(mi*mi) =

2.4 miles\(^2\)

So for any conversion from area to area: if you find the original area in small units, multiply that area by the scale factor squared to get the larger actual area.

Bunuel sidestepped the whole issue by
-- multiplying one length (radius in inches) by one scale factor (2) to get [another] one length: the radius in miles.
Then he found the area in square miles with normal calculations for a circle's area.

Length: multiply by one scale factor \(k\)

Area = length * length, so multiply by two scale factors \(k^2\)

Volume: multiply by three scale factors \(k^3\) (b/c V = L * L * L)

It depends on the problem, but often it is easier to convert one length of the smaller figure to one length of the larger figure (here, radius), and then calculate area or volume.
If you do it that way, you only deal with the scale factor once: length --> length.

Hope that helps.