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The Morphing Dart Board

A version of this problem was found in an old workshop packet from the 1980's. I have taken the liberty of making a few adjustments.

You have a triangle anchored inside a rectangle, with the dimensions shown.

For the following x-values, find the ratio of the area of the triangle to the area of the rectangle.
  • x = 1
  • x = 5
  • x = 10
  • x = 50
  • x = 100
  • x = 500
  • x = 1000
  • x = 5000
  • x = 10000
These ratios actually equal the respective probability of throwing a dart at random into the rectangle and hitting the triangle's interior.

Question: As the value of x continues to increase, what is happening to the ratio? Does it have a limiting value?

Extension Question: Try to visualize this problem to determine the "limit value" without doing any numerical calculations? It is okay to draw a picture....

Note: For those who have studied calculus, this problem seems to nicely illustrate the limit value for a ratio of two quadratics!


Hint: Write a formula for the respective areas of the triangle and the rectangle. Then, substitute values of x to determine the suggested ratios. What do you notice?


Solution Commentary: Behold...