Question: The advice about apples and ranges is not right. Consider the familiar equations for a circle C = 2r and A = r2. (a) Check

The advice about apples and ranges is not right. Consider the familiar equations for a circle C = 2πr and A = πr2.
(a) Check that C and A have different dimensional formulas.
(b) Produce an equation that is not dimensionally homogeneous (i.e., it adds apples and oranges) but is nonetheless true of any circle.
(c) The prior item asks for an equation that is complete but not dimensionally homogeneous. Produce an equation that is dimensionally homogeneous but not complete.
(Just because the old saying isn't strictly right, doesn't keep it from being a useful strategy. Dimensional homogeneity is often used to check the plausibility of equations used in models. For an argument that any complete equation can easily be made dimensionally homogeneous.

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