Questions: The 5 x 5 array of dots in the shown figure represents trees in an orchard. If you were standing at the central spot marked C, you would not be able to see 8 of the 24 trees (shown as X). If you were standing at the center of a 9 x 9 array of trees, how many of the 80 trees would be hidden? (A) 45 (B) 44 (C) 40 (D) 36 (E) 32

The 5 x 5 array of dots in the shown figure represents trees in an orchard. If you were standing at the central spot marked C, you would not be able to see 8 of the 24 trees (shown as X). If you were standing at the center of a 9 x 9 array of trees, how many of the 80 trees would be hidden? (A) 45 (B) 44 (C) 40 (D) 36 (E) 32
Transcript text: The $5 \times 5$ array of dots in the shown figure represents trees in an orchard. If you were standing at the central spot marked $C$, you would not be able to see 8 of the 24 trees (shown as $X$ ). If you were standing at the center of a $9 \times 9$ array of trees, how many of the 80 trees would be hidden? (A) 45 (B) 44 (C) 40 (D) 36 (E) 32
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Solution

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Solution Steps

To solve this problem, we need to understand the pattern of visibility from the center of a grid of trees. In the $5 \times 5$ grid, 8 out of 24 trees are hidden. We can use this ratio to estimate the number of hidden trees in a $9 \times 9$ grid. The number of hidden trees is likely related to the number of lines of sight blocked by other trees. We can calculate the number of hidden trees by considering the increase in the number of rows and columns and applying a similar visibility pattern.

Step 1: Understanding the Problem

We are given a \(9 \times 9\) grid of trees, where we need to determine how many trees are hidden from the center position. In a \(5 \times 5\) grid, it was found that 8 out of 24 trees were hidden.

Step 2: Establishing the Ratio

From the \(5 \times 5\) grid, the ratio of hidden trees to total trees is given by: \[ \text{Hidden Ratio} = \frac{8}{24} = \frac{1}{3} \]

Step 3: Calculating Total Trees in the \(9 \times 9\) Grid

The total number of trees in a \(9 \times 9\) grid is: \[ \text{Total Trees} = 9 \times 9 = 81 \] Since we exclude the center tree, the number of visible trees is: \[ \text{Visible Trees} = 81 - 1 = 80 \]

Step 4: Estimating Hidden Trees

Using the established hidden ratio, we can estimate the number of hidden trees: \[ \text{Hidden Trees} = \text{Hidden Ratio} \times \text{Visible Trees} = \frac{1}{3} \times 80 = \frac{80}{3} \approx 26.67 \] Rounding down, we find that approximately 26 trees are hidden.

Final Answer

Thus, the number of hidden trees in the \(9 \times 9\) grid is \\(\boxed{26}\\).

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