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.
These ratios actually equal the respective probability of throwing a dart at random into the rectangle and hitting the triangle's interior.
- x = 1
- x = 5
- x = 10
- x = 50
- x = 100
- x = 500
- x = 1000
- x = 5000
- x = 10000
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...