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.
After doing this problem, it may be understandable why mathematical relationships were not used.
- 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?
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.