Screening Mammography -- Partial Solution

First notice that the language used in the public service ad is less than 100% clear. Rather than clarifying this in the handout, this is good practice with how things are worded in the real world. So, it was necessary to be able to match up "a lump easily detected by a mammogram" with "Average size of breast cancer determined by mammography" for example.

They also don't tell us what the "size" measurements mean. Radius? Diameter? Other? We need to reason and ask "what makes the most sense?"

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Think about this on your own first..

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For a given sphere (of radius "R" and diameter "D") its volume is definitely NOT (4/3)(π)(D^3). However when you are taking the quotient of the volumes of two spheres, we can use diameters (lets call the second sphere's diameter d and radius r) because:

[(4/3)(π)(D^3)] / [(4/3)(π)(d^3)] 

= [(4/3)(π)((2R)^3)] / [(4/3)(π)((2r)^3)] 

=  [(4/3)(π)(8)(R^3)] / [(4/3)(π)(8)(r^3)]

 = [(4/3)(π)(R^3)] / [(4/3)(π)(r^3)] 

since the 8's cancel.

So this little shortcut lets us skip the "first divide by two on your calculator" step and go right into the cubing etc in the formula, as done in the solution here. Of course the (4/3) and (π) could have been skipped for similar reasons but I did not wish to do so since any relation to sphere volumes would be lost and, worse, it would appear (falsely!) that we are assuming each tumor is a cube since we'd only have the "cube's volume" formula in that case.

Hence the ratio of volumes of the two spheres is equal to

(4/3)(π)(2.8)^3 / (4/3)(π)(1.1)^3 = 
(2.8)^3/(1.1)^3 = (2.8 / 1.1)^3 ≈ 16.45 or close to 17.

Finally, how can be justify more carefully why the ratio of spheres' volumes is a reasonable approximation for the ratio of the number of cells in one tumor versus the number of cells in another?

Ratio of number of cells

= (Number of cells in larger tumor) / (Number of cells in smaller tumor)

≈ratio of masses (Why? What reasonable assumption allows you to say that the quotient of the number of cells in one vs the other tumor, corresponds to the quotient of their mass? Answer: we're assuming "number of cells per unit of mass (e.g. "per gram") is roughly constant)

≈ratio of volumes (Why? Because if we reasonably assume that density is roughly same among all tumors compared, then quotient of volumes will correspond to quotient of masses: the two quotients are equal if density is constant, do you see why?)

Thus the radio of volumes (which we round to be about 16.45) does indeed correspond to the ratio of the number of cells, which is what the Public Service Health ad, claimed was roughly 17.

So (put aside why it was rounded up, perhaps they used a slightly different models or just didn't want to say 'more than 16 times') the "17" is justifiable.