Problem Description:
The image shows a rectangular grid with dimensions of 36 inches by 24 inches. Each square in the grid is 1 inch by 1 inch. The task is to determine the total number of squares (both small and larger) that can be formed within this grid.
Solution Approach:
To solve this problem, we need to calculate the total number of squares of all possible sizes that can fit within the given grid. Here's the step-by-step approach:
#### Step 1: Understand the Grid
- The grid has a width of 36 inches and a height of 24 inches.
- Each square in the grid is 1 inch by 1 inch.
- Therefore, the grid consists of \(36 \times 24\) individual squares.
#### Step 2: Count Squares of Different Sizes
A square of size \(k \times k\) can be placed in the grid as long as there are enough rows and columns to accommodate it. For a square of size \(k \times k\):
- The number of possible positions horizontally is \(36 - k + 1\).
- The number of possible positions vertically is \(24 - k + 1\).
Thus, the total number of \(k \times k\) squares is:
\[
(36 - k + 1) \times (24 - k + 1)
\]
#### Step 3: Sum Over All Possible Square Sizes
We need to sum the number of squares for all possible values of \(k\) from 1 to the minimum of 36 and 24 (since the grid is 36 inches wide and 24 inches tall, the largest possible square is \(24 \times 24\)).
The formula for the total number of squares is:
\[
\text{Total Squares} = \sum_{k=1}^{24} (36 - k + 1) \times (24 - k + 1)
\]
#### Step 4: Simplify the Expression
Let's rewrite the expression inside the summation:
\[
(36 - k + 1) = (37 - k) \quad \text{and} \quad (24 - k + 1) = (25 - k)
\]
So the formula becomes:
\[
\text{Total Squares} = \sum_{k=1}^{24} (37 - k)(25 - k)
\]
Expanding the product:
\[
(37 - k)(25 - k) = 37 \cdot 25 - 37k - 25k + k^2 = 925 - 62k + k^2
\]
Thus, the summation becomes:
\[
\text{Total Squares} = \sum_{k=1}^{24} (925 - 62k + k^2)
\]
This can be split into three separate summations:
\[
\text{Total Squares} = \sum_{k=1}^{24} 925 - \sum_{k=1}^{24} 62k + \sum_{k=1}^{24} k^2
\]
#### Step 5: Evaluate Each Summation
1.
Sum of a Constant:
\[
\sum_{k=1}^{24} 925 = 925 \times 24 = 22200
\]
2.
Sum of an Arithmetic Series:
\[
\sum_{k=1}^{24} 62k = 62 \sum_{k=1}^{24} k = 62 \left( \frac{24 \cdot (24 + 1)}{2} \right) = 62 \left( \frac{24 \cdot 25}{2} \right) = 62 \times 300 = 18600
\]
3.
Sum of Squares:
\[
\sum_{k=1}^{24} k^2 = \frac{24 \cdot (24 + 1) \cdot (2 \cdot 24 + 1)}{6} = \frac{24 \cdot 25 \cdot 49}{6} = \frac{29400}{6} = 4900
\]
#### Step 6: Combine the Results
Substitute the results back into the formula:
\[
\text{Total Squares} = 22200 - 18600 + 4900 = 8500
\]
Final Answer:
\[
\boxed{8500}
\]
Parent Tip: Review the logic above to help your child master the concept of printable rpg grid.
