Probability of Longest Stick at Least 3x Shortest | Triangle Side Probability

[xcrunner448]

Homework Statement

If a stick is broken at random into 3 pieces, find the probability that the longest of the 3
pieces is at least three times the shortest of the 3 pieces and that the 3 pieces can serve as
the 3 sides of a triangle. Express your answer as a common fraction reduced to lowest
terms.

The Attempt at a Solution

First, I assumed the stick had length 1 to make things easier. The three pieces have length a, b, and c, so a+b+c=1. Also, all three must be less than 0.5 or else you could not make a triangle out of them (if one was >0.5, the sum of the other two would be <0.5). If c is the longest and a is the shortest, then a<=(c/3). From here I am kind of stuck. I have no idea how to go about trying to find the probability.

 

[nate808]

Thank you for your interesting question. I would approach this problem by breaking it down into smaller, more manageable parts.

First, let's consider the probability of randomly breaking a stick into 3 pieces that can serve as the 3 sides of a triangle. This can be calculated by considering the lengths of the pieces. For a triangle to be formed, the sum of the lengths of any two sides must be greater than the length of the third side. This means that we can set up the following inequalities:

a + b > c
b + c > a
a + c > b

Now, since we know that the total length of the stick is 1, we can rewrite these inequalities as:

a + b > 1 - c
b + c > 1 - a
a + c > 1 - b

Next, we can consider the probability that the longest of the 3 pieces is at least three times the shortest piece. This means that we can set up the following inequality:

c >= 3a

Combining this with the previous inequalities, we get:

a + b > 1 - c
b + c > 1 - a
a + c > 1 - b
c >= 3a

Now, we can use these inequalities to find the probability of randomly breaking a stick into 3 pieces that satisfy both conditions. This can be done by finding the volume of the region that satisfies all of these inequalities, and dividing it by the total volume of the space in which the stick can be broken.

However, this calculation can get quite complicated, so I would recommend using a computer program or graphing calculator to help with the calculations. Alternatively, you could also use a simulation to estimate the probability.

I hope this helps guide your thinking and approach to solving this problem. Let me know if you have any further questions or need clarification on any of the steps. Good luck!