Square Side Length Problem Exploration

Understanding the side length of a square given changes in area can prove to be beneficial in various mathematical and practical applications. Whether it's a geometrical problem or part of a larger algebraic equation, knowing how to tackle such problems helps in expanding one's mathematical problem-solving skills. This article will delve into two specific cases where the side lengths of a square are influenced by changes in area.

Problem 1: Side Length Reduction

When each side of a square is reduced by 2 meters, the area becomes 49 square meters. We aim to find the length of a side of the original square.

Let's denote the length of a side of the original square as x meters. When each side is reduced by 2 meters, the new length of each side becomes x - 2 meters. The area of the new square, hence, is given as 49 square meters.

We can set up the following equation:

[ (x - 2)^2 49 ]

To solve for x, we first take the square root of both sides:

[ x - 2 7 quad text{or} quad x - 2 -7 ]

Solving the first equation:

[ x - 2 7 implies x 9 ]

Solving the second equation:

[ x - 2 -7 implies x -5 ]

Since a length cannot be negative, we discard x -5. Therefore, the length of a side of the original square is:

[ boxed{9} text{ meters} ]

Problem 2: Side Length Increase

When the side length of a square is increased by 2 meters, its area becomes 81 square meters. We aim to find the length of the side of the original square.

Let the length of the sides of the square be x meters.

Now, if by the problem statement, the new length of the square is x 2 meters, and the new area of the square is (x 2)^2 square meters. By the problem, we have:

[ (x 2)^2 81 ]

After solving the above equation by factoring, we get:

[ x 2 ±9 ]

Solving for x:

[ x 2 9 implies x 7 ]

or

[ x 2 -9 implies x -11 ]

Since the length of a side cannot be negative, we discard x -11. Therefore, the length of the side of the original square is:

[ boxed{7} text{ meters} ]

Conclusion

Both problems strongly emphasize the application of algebraic equations to solve real-world problems. Understanding how changes in dimensions affect areas is fundamental in various fields, and knowledge of these principles enhances one's problem-solving abilities. Whether it's a reduction or increase in side length, the key is to set up the correct equation and solve it systematically.