Question 1199950: Find the maximum number of non-overlapping regions into which 10 additional straight lines can subdivide this figure.
Figure: https://ibb.co/99bk08m
Thanks so much!
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **1. Initial Regions:**
* The rectangle is already divided into 4 smaller rectangles.
**2. Effect of Each Line:**
* **First Line:** At most, it can intersect existing lines twice, creating 2 new regions.
* **Second Line:** At most, it can intersect the first line and existing lines twice, creating 3 new regions.
* **Third Line:** At most, it can intersect the first two lines and existing lines twice, creating 4 new regions.
* **And so on...**
**3. General Pattern:**
* Each subsequent line can potentially intersect all the previous lines at most once, creating one additional region for each intersection.
**4. Maximum Regions:**
* **Initial Regions:** 4
* **First Line:** 4 + 2 = 6 regions
* **Second Line:** 6 + 3 = 9 regions
* **Third Line:** 9 + 4 = 13 regions
* **Fourth Line:** 13 + 5 = 18 regions
* **Fifth Line:** 18 + 6 = 24 regions
* **Sixth Line:** 24 + 7 = 31 regions
* **Seventh Line:** 31 + 8 = 39 regions
* **Eighth Line:** 39 + 9 = 48 regions
* **Ninth Line:** 48 + 10 = 58 regions
* **Tenth Line:** 58 + 11 = 69 regions
**Therefore, the maximum number of non-overlapping regions into which 10 additional straight lines can subdivide the figure is 69.**
**Note:** This assumes that the lines are strategically placed to maximize the number of intersections and, consequently, the number of regions.
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