To determine how many different triangles Joanna can form with her four straws, we need to consider the lengths of the straws and the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let’s name Joanna’s straws based on their lengths: Straw A (length 3), Straw B (length 4), Straw C (length 5), and Straw D (length 6). Joanna can choose any three straws out of the four to attempt to form a triangle. The possible combinations of three straws from four can be calculated using the combination formula (4 choose 3), which equals four combinations.
These combinations are:
- Straws A, B, and C
- Straws A, B, and D
- Straws A, C, and D
- Straws B, C, and D
Next, we evaluate each combination using the triangle inequality theorem. For the combination of Straws A, B, and C (3, 4, 5), the sums of any two sides (3+4=7, 3+5=8, 4+5=9) are greater than the remaining side, so they can form a triangle.
For Straws A, B, and D (3, 4, 6), the sums of two sides (3+4=7, 3+6=9, 4+6=10) also satisfy the triangle inequality. Straw combinations A, C, and D (3, 5, 6) (3+5=8, 3+6=9, 5+6=11) can form a triangle as well. Finally, for Straws B, C, and D (4, 5, 6), the sums of two sides (4+5=9, 4+6=10, 5+6=11) also comply with the triangle inequality.
In conclusion, Joanna can make a total of four different triangles from her combinations of straws: using Straws A, B, and C; A, B, and D; A, C, and D; and B, C, and D. All combinations satisfy the necessary conditions to form triangles.