5. You are given a bag filled with squares made out of wire. No two squares have the same size. The bag contains a

single square of length equal to every possible integer between 37 and 73 inclusive. You must discard from the bag

only those squares that have an odd area.

(a) What the the total length of wire used to make all the remaining squares in the bag?

(b) Write a pseudo code algorithm entitled SUM_SQUARES that solves this problem using a loop.

### Problem Analysis

1. **Squares in the bag**: The bag contains squares made of wire with side lengths ranging from 37 to 73 (inclusive).
2. **Areas to discard**: You must discard squares that have an odd area. Since the area of a square is given by \( \text{side}^2 \), the area will be odd if and only if the side length is odd.
3. **Remaining squares**: We need to find the total wire length used for the remaining squares.

### Solution Steps

1. **Identify Remaining Squares**:
   - Only keep squares with even side lengths because their areas will be even.

2. **Calculate the Perimeter**:
   - The perimeter of each square is \( 4 \times \text{side length} \).

3. **Compute the Total Wire Length**:
   - Sum the perimeters of the remaining squares.

### Calculation

Let's compute the total wire length using the steps described.

1. **Range of side lengths**: 37 to 73.
2. **Squares with even side lengths**: From 38 to 72 (even numbers within the range).

We'll compute the total wire length using these side lengths.**(a) The total length of wire used to make all the remaining squares in the bag is 3960 units.**

### (b) Pseudocode Algorithm: `SUM_SQUARES`

Here is the pseudocode to solve this problem using a loop:

```plaintext
ALGORITHM SUM_SQUARES
    INPUT: None
    OUTPUT: Total length of wire used for the remaining squares

    # Initialize total wire length to 0
    total_wire_length ← 0

    # Loop through each side length from 37 to 73 inclusive
    FOR side_length FROM 37 TO 73 DO
        # Check if the side length is even (even area)
        IF side_length MOD 2 = 0 THEN
            # Calculate perimeter and add to the total wire length
            total_wire_length ← total_wire_length + (4 * side_length)
        END IF
    END FOR

    # Output the total wire length
    RETURN total_wire_length
END ALGORITHM
```

This algorithm iterates through all side lengths, discards those with odd areas, and accumulates the perimeter of the remaining squares.

Would you like further clarifications, or do you have any other questions? Here are related explorations:

1. How does the condition of side lengths affect the area?
2. What is the significance of even and odd numbers in mathematical problems?
3. How would you optimize this algorithm for larger ranges?
4. How can you generalize this problem for other shapes, like triangles?
5. What are some real-world applications of algorithms like this?

**Tip**: Understanding how to filter elements using conditions is crucial in programming, as it helps in solving problems efficiently!

Learn how to calculate the total length of wire used to make squares with even areas from a bag containing squares with side lengths ranging from 37 to 73. This problem involves identifying squares with even side lengths and computing their perimeters. Suitable for Grades 7-9.

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