possible range of line AB

Question

The line segment AB is 15 inches long and is to be cut into five smaller pieces. Each of the smaller pieces has to be greater than 0 inches in length.

Depending on how the segment AB is cut, the range vari- es. What are the possible values for the range? Answer with an interval ... Ive been working with this, My interval is [a, b] where a is zero (0) because that will be the least if the case is all 5 segments are equal. I dont know what should be the value of b, im thinking that it could be 15-4x where x is the least possible value of first segment, and considering if what if 4 of them are equal but i still dont know what it is because there are infinitely many of them that approach to zero...

Answer 1

Comment continued: In my initial comment I suggested some limits. Just for fun, I simulated randomly cutting a million pieces of string, each of length 15, in this way to see how close I could get to the theoretical maximum and minimum values for the range. The answers were about 0.21 for the minimum and about 14.8 for the maximum. So it seems unusual with random cuts to get ranges really near the endpoints of the open interval $(0,15).$

m = 10^5;
r = replicate(m, diff(range(diff(sort(c(0,runif(4,0,15),15))))) )
min(r); max(r)
## 0.2062757    # larger than 0
## 14.80289     # smaller than 15

A histogram of the one million simulated ranges for random cuts is shown below:

Note: The simulation was done in R. A quirk of R is that the range returns the sample minimum and maximum, so if you want the the actual range you have to take the difference of the two values returned. Simulation details for one string:

ends = c(0, runif(4, 0, 15), 15);  ends
##  0.00000 10.50323 11.64744 10.80809 14.53096 15.00000
sort(ends)
##  0.00000 10.50323 10.80809 11.64744 14.53096 15.00000
lengths = diff(sort(ends));  lengths
## 10.5032309  0.3048597  0.8393522  2.8835177  0.4690396
max.min = range(lengths);  max.min
##  0.3048597 10.5032309
range = diff(max.min);  range
## 10.19837