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Making Sense of Cents

If only our forefathers and foremothers had been wise enough to construct money that followed some mathematical relationship. Why is it that a dime is smaller than a penny, yet a penny is smaller than a nickel? At least the quarter and half-dollar are greater than a nickel. And who can remember the size of the dollar coin!

So, here is the task. You are given two facts: A nickel is 21.21 mm. in diameter and a dime is 17.91 mm. in diameter.

Now, suppose the size of money was constructed according to a linear functional relationship f(x), where x is value of the coin in cents and f(x) is the diameter of the coin in mm.

  • What should be the diameters of a penny, a quarter, and a half-dollar?
  • How do your answers change if x was the area of the coin?
  • And in both cases of x (diameter and area), how do your answers change if the assumed functional relationship was exponential rather than linear?
After doing this problem, it may be understandable why mathematical relationships were not used.

NOTE: Andy D., who teaches in Singapore, has added the information that in Australia, the size of the coin increases with its value....but it is not a linear relationship. Anyone want to do some curve-fitting and determine the relationship...focus on the pre-decimal coins only?

 

Source: MS ad


Hint: You are given two points...find the slope and y-intercept.

 


Solution Commentary: When x is the diameter, f(x)= -0.66x + 24.51. This implies that a penny should have a diameter of 23.85 mm., a quarter should have a diameter of 8.01 mm., and a half-dollar should have a diameter of -8.49 mm.! Oops....

The cases of both x as the area and the exponential relationship will be left for you. The nice thing is that at least the latter prevents negative diameters.