Solving the Geometry Puzzle: Calculating the Length and Width of a Rectangular Garden

This article is an exploration of a fun and challenging geometry problem. We will delve into solving the puzzle of finding the length and width of a rectangular garden using the ancient Pythagorean theorem. This problem is not only intriguing but also a great way to practice and apply fundamental mathematical principles.

The Problem Statement

The problem states that the length of a rectangular garden is 3 units greater than its width. If the diagonal of the garden measures 15 units, what is its length?

Mathematical Approach Using Pythagoras' Theorem

Let's denote the width of the garden as (w). According to the problem, the length of the garden can be expressed as (w 3). The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the other two sides. Applying this to our garden, we have:

(15^2 w^2 (w 3)^2)

Expanding and simplifying this equation:

(225 w^2 (w^2 6w 9))

(225 2w^2 6w 9)

Bringing all terms to one side:

(2w^2 6w 9 - 225 0)

(2w^2 6w - 216 0)

Dividing the entire equation by 2 to simplify:

(w^2 3w - 108 0)

We can solve this quadratic equation using the quadratic formula, (w frac{-b pm sqrt{b^2 - 4ac}}{2a}), where (a 1, b 3, c -108). Substituting these values in:

(w frac{-3 pm sqrt{3^2 - 4(1)(-108)}}{2(1)})

(w frac{-3 pm sqrt{9 432}}{2})

(w frac{-3 pm sqrt{441}}{2})

(w frac{-3 pm 21}{2})

This gives us two potential solutions:

(w frac{18}{2} 9)

(w frac{-24}{2} -12)

Since the width cannot be a negative value, we take (w 9). Therefore, the width of the garden is 9 units, and the length (which is 3 units more) is:

(9 3 12)

So, the length of the garden is 12 units.

Verification of the Solution

To verify, we can check that the Pythagorean theorem holds:

(9^2 12^2 81 144 225 15^2)

This confirms our solution is correct. The area of the garden can be calculated as:

(A w times l 9 times 12 108 , text{square units})

Conclusion

In conclusion, the length of the rectangular garden is 12 units, given that the width is 9 units and the diagonal is 15 units. This problem effectively demonstrates the application of the Pythagorean theorem in solving real-world geometry puzzles.

Key Takeaways:

The Pythagorean theorem is a powerful tool for solving geometric problems. Quadratic equations can be solved using the quadratic formula. Real-world applications can be solved with mathematical principles.