Solving for the Dimensions of a Garden: A Real-World Application of Quadratics

In this article, we will go through a practical problem involving the calculation of the dimensions of a garden. We will apply the concept of quadratic equations to solve the problem step by step. This example will not only demonstrate the theoretical use of quadratic equations but also highlight their real-world applicability.

Scenario: Finding the Dimensions of a Garden

Suppose we have a garden with an area of 90 square meters. Furthermore, the problem states that the length of the garden is 3 meters more than twice the width. To find the dimensions of the garden, we need to set up and solve a quadratic equation.

Setting Up the Equation

Let's denote the width of the garden by ( w ) meters. According to the problem, the length of the garden ( l ) can be expressed as:

[ l 2w 3 ]

The area ( A ) of the garden is given by the product of its length and width:

[ A l times w ]

Given that the area is 90 square meters, we can substitute the expression for length into the area formula:

[ 90 (2w 3) times w ]

Solving the Quadratic Equation

Let's expand and rearrange the equation to solve for ( w ):

[ 90 2w^2 3w ]

Rewriting it in the standard form of a quadratic equation, we get:

[ 2w^2 3w - 90 0 ]

Using the quadratic formula, which is:

[ w frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

We have:

( a 2 ) ( b 3 ) ( c -90 )

First, we calculate the discriminant:

[ b^2 - 4ac 3^2 - 4 times 2 times (-90) 9 720 729 ]

Now, we can apply the quadratic formula:

[ w frac{-3 pm sqrt{729}}{2 times 2} ]

Calculating the two possible values for ( w ), we get:

[ w frac{24}{4} 6 ]

[ w frac{-30}{4} -7.5 quad text{(not a valid solution since width cannot be negative)} ]

Therefore, the width of the garden is ( w 6 ) meters. Now we can find the length:

[ l 2w 3 2(6) 3 15 text{ meters} ]

Summary of Dimensions

The dimensions of the garden are:

Width: 6 meters Length: 15 meters

In conclusion, the problem was successfully solved using the quadratic equation, demonstrating its applicability in real-world scenarios like determining the dimensions of a garden.

Mathematical Insight

While the above problem is straightforward, it provides a fundamental understanding of how quadratic equations can be used to find unknown quantities in a geometric context. The process involves setting up the equation, solving for the unknowns, and then interpreting the results in the context of the problem.

Real-World Applications

Quadratic equations are not just abstract mathematical concepts but have numerous practical applications. They can be used in various fields such as engineering, physics, and economics. For example, in engineering, quadratic equations are used to analyze the trajectories of objects, optimize the design of structures, and model the behavior of various systems. In physics, they are used to calculate the motion of projectiles and the behavior of electrical circuits.

Conclusion

The dimensions of the garden were successfully found using the quadratic equation method. This example highlights the importance of quadratic equations in solving real-world problems and how they can be applied in practical situations. Understanding these concepts can be crucial for students and professionals in various fields, providing them with the tools to solve complex problems and make informed decisions.