Welcome to the tree planting problem! This is a classic mathematical application problem that studies the relationship between the number of objects, like trees, and the intervals between them. We can see here a line segment with five trees planted at equal intervals, creating four spaces between them.
When planting trees on a line segment, we have three different cases. First, if we plant trees at both ends, the number of trees equals the number of intervals plus one. Second, if we plant at only one end, trees equal intervals. Third, if we don't plant at either end, trees equal intervals minus one.
When planting trees around closed figures like circles or squares, the relationship becomes simpler. The number of trees always equals the number of intervals. This is because in a closed shape, there are no endpoints, so every tree creates exactly one interval between itself and the next tree.
Let's solve a practical example. We need to plant trees along a 100-meter road with 10-meter intervals. First, we calculate the number of intervals: 100 divided by 10 equals 10 intervals. Since we're planting at both ends of the road, the number of trees equals intervals plus one, which gives us 11 trees total.
To summarize what we've learned about tree planting problems: These problems help us understand the relationship between objects and the intervals between them. For line segments, we must consider whether trees are planted at the endpoints. For closed figures, the number of trees always equals the number of intervals. These mathematical principles can be applied to many real-world spacing and arrangement problems.