Discussion
A proportional circle representing 100 units has a radius of 4 mm. If area is scaled in proportion to the value, what radius represents 400 units?
Answer: C
Correct answer: 8 mm
Area is scaled with the value, and area goes with the square of the radius, so the radius goes with the square root of the value: \[ r = 4\sqrt{\frac{400}{100}} = 4\sqrt{4} = 8\ \mathrm{mm}. \] Scaling the radius directly by the fourfold increase gives the 16 mm offered, which would make the circle sixteen times the area rather than four - the classic exaggeration in proportional symbol work.
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