Garden Dimensions: Calculating Length & Width With A Tomato Patch

Hey guys! Let's dive into a fun math problem about planning a vegetable garden. This is something that combines a little bit of algebra with real-world gardening, and it's a great way to see how math can help us in everyday life. We're going to break down the problem step by step, so you can follow along and understand how to solve it. Our main focus is on figuring out the dimensions – the length and the width – of a rectangular garden, especially when there's a square patch for tomatoes involved. Understanding these calculations is crucial for any gardener who wants to maximize space and yield.

Setting Up the Scenario

Okay, so picture this: Melissa is dreaming of a lush vegetable garden. She's got all sorts of ideas, but she wants to plan it out perfectly before she starts digging. The main shape of her garden will be a rectangle, which is pretty standard for most gardens. But here's the twist – within this rectangle, she wants a special square-shaped area just for her tomatoes. Tomatoes need lots of sunshine and care, so giving them their own dedicated space is a smart move. This setup isn't just about planting veggies; it's about optimizing space and ensuring each plant thrives. The square tomato patch is central to the garden's design, influencing the dimensions of the entire rectangular space.

Defining the Variables

Before we jump into the math, let's get our terms straight. In algebra, we often use letters to represent unknown quantities, and that's exactly what we'll do here. Let's say the side length of the square tomato patch is "x" feet. Since it's a square, all sides are equal, so the width and length are both x. Now, the problem tells us some interesting things about how the rest of the garden relates to this tomato patch. We're particularly interested in how the length and width of the rectangular garden tie into the size of this tomato square. By defining our variables clearly, we set the stage for creating equations that accurately model the garden's layout. This is a critical step in translating a real-world problem into a mathematical one.

Length of the Garden

Here's where it gets interesting. Melissa wants the length of her entire garden to be more than just a little bit longer than the tomato patch. Specifically, she wants the length to exceed three times the length of the tomato patch (x) by 2 feet. So, if the tomato patch is x feet long, three times that is 3x, and adding 2 feet gives us 3x + 2. This expression represents the total length of the rectangular garden. Understanding this relationship is key to visualizing how the tomato patch influences the overall size and shape of the garden. It's a perfect example of how algebraic expressions can capture real-world spatial relationships.

Width of the Garden

Now, let's think about the width. Melissa has a similar idea in mind for the width of the garden. She wants the width to exceed the width of the tomato patch by 5 feet. This is a bit simpler than the length calculation. Since the tomato patch is x feet wide, the garden's width will be x + 5 feet. So, we now have expressions for both the length (3x + 2) and the width (x + 5) of the garden, all in terms of x, the side length of the tomato patch. This connection between the tomato patch and the overall garden dimensions highlights the importance of careful planning in garden design.

Formulating the Equations

Alright, we've got our variables and expressions. Now comes the fun part: turning this into a mathematical equation or equations. The beauty of algebra is that it lets us take a word problem and translate it into a language we can solve. In this case, we've already established expressions for the length and width of the garden. However, to find specific values for these dimensions, we might need additional information or constraints. The problem as stated primarily sets up the relationships between the tomato patch and the garden's dimensions, rather than giving us a direct equation to solve for x. To proceed further, we might need information like the total area of the garden, or a limit on the amount of fencing Melissa wants to use. Without additional constraints, we have a set of expressions that describe the garden's dimensions in terms of the tomato patch size, but not a single definitive solution. Therefore, the next step would typically involve introducing additional information or a specific goal (like minimizing cost or maximizing area) to create a solvable equation. The relationships we've defined so far are crucial, but they're just the starting point. The real power of algebraic problem-solving comes into play when we can combine these relationships with specific conditions to find concrete answers.

Understanding the Relationships

Before we go any further, let's pause and make sure we really understand the relationships we've set up. The length of the garden (3x + 2) is directly tied to the size of the tomato patch (x). The bigger the tomato patch, the longer the garden needs to be. Similarly, the width of the garden (x + 5) also depends on x. These aren't just random numbers; they're carefully planned dimensions that Melissa is thinking about. Visualizing these relationships is key to solving this kind of problem. Imagine the tomato patch growing – as it gets larger, the whole garden expands in both length and width. This is a great way to internalize how the variables interact. By grasping these connections intuitively, we can better approach the mathematical challenges ahead.

The Need for More Information

Okay, guys, here's a crucial point. As it stands, we can't solve for a specific value of x (the side length of the tomato patch). We have expressions for the length and width of the garden, but we don't have an equation that lets us pin down a numerical value for x. Think of it like this: we know how the ingredients in a recipe relate to each other, but we don't know the total amount we want to make. To actually bake something, we need to know the final quantity. In our garden problem, we need more information, like the total area of the garden, the amount of fencing Melissa wants to use, or perhaps a desired ratio between the area of the tomato patch and the rest of the garden. Without these additional constraints, we can explore different possibilities for the garden's dimensions, but we can't arrive at a single, definitive answer. This highlights a fundamental aspect of problem-solving: identifying what information is missing and understanding how that information would help us reach a solution.

Solving with an Example Scenario

Let's make things more concrete by adding some information. Suppose Melissa wants the total area of her garden to be 100 square feet. Now we have a constraint that we can use to create an equation and solve for x. This is where the problem shifts from simply setting up relationships to actually finding a solution. Introducing a specific area requirement gives us a target – a fixed amount of space that Melissa wants to work with. This is a very practical consideration in garden planning, as it might be dictated by the size of her yard, the amount of sunlight available, or the amount of time she wants to spend gardening. By integrating this area constraint, we transform the problem into a more typical algebraic puzzle that we can solve systematically.

Setting Up the Area Equation

We know that the area of a rectangle is length times width. We also know the length is (3x + 2) and the width is (x + 5). And now we know the total area is 100 square feet. So, we can write the equation: (3x + 2)(x + 5) = 100. This equation is the key to solving the problem. It combines our expressions for length and width with the total area constraint to create a mathematical statement that we can manipulate to find the value of x. This is a classic example of how algebra allows us to translate a real-world scenario into a symbolic representation that we can analyze and solve. The equation captures the essence of the problem: the dimensions of the garden must be such that their product equals the desired total area.

Expanding and Simplifying

Our next step is to expand and simplify the equation. When we multiply (3x + 2) by (x + 5), we get 3x² + 15x + 2x + 10, which simplifies to 3x² + 17x + 10. So, our equation becomes 3x² + 17x + 10 = 100. Now, to solve this quadratic equation, we need to set it equal to zero. We do this by subtracting 100 from both sides, giving us 3x² + 17x - 90 = 0. This transformation is crucial because it puts the equation into a standard form that we can solve using various techniques, such as factoring, completing the square, or using the quadratic formula. By simplifying the equation, we make it more manageable and unlock the potential to find the value(s) of x that satisfy the area constraint.

Solving the Quadratic Equation

Now we have a quadratic equation: 3x² + 17x - 90 = 0. There are a few ways to solve this, but let's use the quadratic formula. The quadratic formula is a powerful tool that allows us to find the solutions to any quadratic equation in the form ax² + bx + c = 0. It's a reliable method, especially when factoring is difficult or impossible. Applying the quadratic formula involves identifying the coefficients a, b, and c, and then plugging them into the formula. This is a straightforward process, but it's important to be careful with the calculations to avoid errors. The quadratic formula is a cornerstone of algebra, and mastering its use is essential for solving a wide range of mathematical problems.

Applying the Quadratic Formula

The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a). In our equation, a = 3, b = 17, and c = -90. Plugging these values in, we get x = (-17 ± √(17² - 4 * 3 * -90)) / (2 * 3). Let's break this down: 17² is 289, and 4 * 3 * -90 is -1080. So, inside the square root, we have 289 - (-1080), which is 289 + 1080 = 1369. The square root of 1369 is 37. This simplifies our equation to x = (-17 ± 37) / 6. We now have two possible solutions for x. This process of carefully substituting values into the formula and simplifying step by step is a testament to the precision required in algebra. Each operation must be performed accurately to arrive at the correct solutions.

Finding the Two Possible Solutions

So, we have two possibilities for x: x = (-17 + 37) / 6 and x = (-17 - 37) / 6. Let's calculate them. The first one is x = 20 / 6, which simplifies to approximately 3.33 feet. The second one is x = -54 / 6, which is -9 feet. Now, here's where we need to use our common sense. Can the side of a tomato patch be negative? Nope! So, we discard the negative solution. This is a critical step in problem-solving: interpreting the mathematical solutions in the context of the real-world scenario. Not all mathematical answers make sense in practice. The negative solution, while mathematically valid, is physically impossible in the context of the garden dimensions. Therefore, we focus on the positive solution, which represents a feasible size for the tomato patch.

Choosing the Realistic Solution

We're left with x ≈ 3.33 feet as the side length of the tomato patch. This makes sense in the real world. Now that we have a realistic value for x, we can calculate the length and width of the entire garden. This is the culmination of our algebraic efforts: we've translated a word problem into an equation, solved the equation, and now we're interpreting the solution in the context of Melissa's garden. The side length of the tomato patch is a key dimension, and with this value, we can determine the other dimensions of the garden and ensure that they fit within Melissa's constraints and preferences.

Calculating the Garden Dimensions

With x approximately 3.33 feet, we can find the length and width of the garden. The length is 3x + 2, which is 3 * 3.33 + 2 ≈ 11.99 feet. We can round that up to 12 feet for simplicity in the real world. The width is x + 5, which is 3.33 + 5 = 8.33 feet. So, Melissa's garden will be about 12 feet long and 8.33 feet wide. This is a practical and useful result. We've gone from abstract algebraic expressions to concrete dimensions that Melissa can use to plan her garden layout. These dimensions will guide her as she prepares the soil, builds the garden bed, and plants her vegetables. The mathematical solution has provided a tangible outcome for a real-world gardening project.

Final Dimensions

So, to recap, the tomato patch will be approximately 3.33 feet by 3.33 feet, the garden will be about 12 feet long, and about 8.33 feet wide. Melissa now has a solid plan for her garden! This process highlights the power of math in everyday planning. By using algebra, we were able to optimize the garden's dimensions, ensuring that Melissa's tomato patch and the rest of her vegetables have the space they need to thrive. This is a practical application of mathematical concepts that demonstrates their relevance beyond the classroom. Melissa's garden plan is a testament to the utility of algebra in real-world situations.

Verifying the Solution

It's always a good idea to double-check our work. Let's calculate the area of the garden using our calculated dimensions: 12 feet * 8.33 feet ≈ 99.96 square feet, which is very close to our target of 100 square feet. This gives us confidence that our solution is accurate. Verifying the solution is a crucial step in the problem-solving process. It ensures that our calculations are correct and that our answers make sense in the context of the problem. In this case, the calculated area is very close to the target area, which confirms that our dimensions for the garden are reasonable and consistent with the given constraints. This final check provides peace of mind and reinforces the reliability of our mathematical approach.

Conclusion

See, guys? Math isn't just about numbers and equations; it's a tool that can help us plan and build things in the real world! This problem showed us how algebra can be used to design a garden, but the same principles can be applied to many other situations, from building furniture to planning a room layout. The key takeaway is that mathematical problem-solving involves not just manipulating symbols, but also understanding the underlying relationships and constraints in a real-world scenario. By translating a practical problem into an algebraic model, we can use the power of mathematics to find optimal solutions and make informed decisions. This is the essence of applied mathematics – using abstract concepts to solve concrete problems.