Hard Maths question

Question

A piece of wood of length 10 cm is to be divided into 3 pieces so that the length of each piece is a whole number of cm, for example 2 cm, 3 cm and 5 cm a) List all the different sets of lengths which could be obtained. b) If one of these sets is selected at random, what is the probability that the lengths of the pieces could be lengths of the sides of a triangle?

Answer 1

We need a system to break 10 into three lengths and see if we can make a triangle from the sides.

We need to break it in the following fashion

A B C triangle

8 1 1 No. 8 > 1 + 1

7 1 2 No. 7 > 1 + 2

6 1 3 No. 6 > 1 + 3

6 2 2 No. 6> 2 + 2

5 1 4 No. 5 = 1 + 4

5 2 3 No. 5 = 2 + 3

4 1 5 No. 5 = 4 + 1 - side C * repeats (5, 1, 4)

4 2 4 Yes 4 < 2+4

4 3 3 Yes 4 < 3+3

3 1 6 * same as above 6, 1, 3.

Note the system I used to make sure all different sets are listed. Two of the sets repeat, so we need to disregard them. We can divide 10 into three pieces in 8 different ways. Note, 8, 1, 1 is the same set as 1, 8, 1.

I think you very easily can calculate the probability to construct a triangle with such a set - pretty low. If one of the pieces is 5 cm or more, we cannot construct a triangle - this piece will dominate the other two.

We start having a triangle from the pieces, when the length A (think of it as the longest side of the triangle), is less than 5. We have only couple of ways we can cut that make a triangle 4 , 2, 4 and 4, 3, 3.