The model builder has 4 pieces of balsa wood that are 4 cm,5 cm, 6 cm, and 7 cm in length. How many different combinations of 3 pieces can be used to make triangles without breaking or cutting the pieces? List the combinations as inequalities.
To determine the number of different combinations of 3 pieces to make triangles without breaking or cutting the pieces, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Let's go step by step to find the combinations:
1. Start with the first piece of balsa wood, which is 4 cm in length. Any other piece selected must have a combined length greater than 4 cm, as per the triangle inequality theorem.
- 4 cm + 5 cm > 6 cm (since 4 + 5 = 9 > 6)
- 4 cm + 6 cm > 7 cm (since 4 + 6 = 10 > 7)
- 4 cm + 7 cm > 5 cm (since 4 + 7 = 11 > 5)
So, there are three valid combinations using the 4 cm piece.
2. Move to the second piece of balsa wood, which is 5 cm in length. Again, any other piece selected must have a combined length greater than 5 cm based on the triangle inequality theorem.
- 5 cm + 6 cm > 7 cm (since 5 + 6 = 11 > 7)
There is only one valid combination using the 5 cm piece.
3. Proceed to the third piece of balsa wood, which is 6 cm in length.
There are no further combinations possible with the 6 cm piece since it is the longest piece.
Therefore, the valid combinations of 3 pieces to make triangles without breaking or cutting the pieces are:
- 4 cm, 5 cm, and 6 cm
- 4 cm, 6 cm, and 7 cm
- 4 cm, 5 cm, and 7 cm
Note: We can write the combinations as inequalities by using the triangle inequality theorem. For example, the first combination can be expressed as:
4 cm + 5 cm > 6 cm
Similarly, the other combinations can be written in the same format.