Tomato Patch Length: Equation For 170 Sq Ft Garden

Hey guys! Let's dive into a fun math problem today! We're tackling a question about finding the length of a tomato patch within a vegetable garden. Specifically, we're given that the total area of the vegetable garden is 170 square feet, and we need to figure out which equation can help us determine the length of our tomato patch. Sounds like a real-world problem, right? Let's break it down step by step. This involves a bit of algebra and spatial reasoning, but don't worry, we'll make it super clear and easy to follow. So, grab your thinking caps, and let's get started!

Understanding the Problem Setup

Okay, so first things first, let's visualize what we're dealing with. Imagine a vegetable garden that has a total area of 170 square feet. Within this garden, there's a specific section dedicated to growing tomatoes, which we're calling the tomato patch. The goal here is to find an equation that relates the dimensions of this tomato patch (specifically its length) to the total area of the garden. Now, you might be wondering, why do we need an equation? Well, equations are just mathematical tools that help us represent relationships between different quantities. In this case, the quantities are the dimensions of the tomato patch and the area of the garden.

To get a better handle on this, let's think about what information we might need to set up such an equation. We'll definitely need to know something about the shape and dimensions of the tomato patch. For example, is it a rectangle? A square? How are its length and width related? We'll also need to understand how the area of the tomato patch contributes to the overall area of the garden. Are there other sections in the garden besides the tomato patch? All these details will play a crucial role in formulating the correct equation. So, stay with me, and we'll uncover these details as we dissect the problem further. Understanding this setup is key to solving this problem, and once we've got a clear picture in our minds, the rest will fall into place much more easily.

Analyzing the Given Options

Alright, now that we've got a solid grasp of the problem's setup, let's take a look at the answer options provided. We have four equations, and our task is to figure out which one accurately represents the relationship between the tomato patch's length and the garden's total area. Remember, the correct equation should help us find the length of the tomato patch when the area of the garden is 170 square feet. Let's list the options here for easy reference:

A. $0=3 x^2+2 x+180$ B. $0=3 x^2-160$ C. $0=3 x^2+10 x+180$ D. $0=3 x^2+17 x-160$

Each of these equations is a quadratic equation, which means it involves a variable (in this case, 'x') raised to the power of 2. Quadratic equations often describe situations involving areas, so it makes sense that we're seeing them here. However, not all quadratic equations are created equal! We need to find the one that correctly models our specific scenario.

To do this, we'll need to think about what the 'x' in each equation represents. Most likely, 'x' is related to the length (or potentially the width) of the tomato patch. We'll also need to consider the other terms in the equation, like the coefficients (the numbers multiplying the 'x' terms) and the constant terms (the numbers without any 'x'). These terms likely represent other dimensions or factors that contribute to the overall area. The key here is to carefully analyze each equation and see if it logically fits the problem's description. We're essentially playing detective, using the clues in the equations to deduce the correct answer.

Determining the Correct Equation (Step-by-Step)

Okay, let's put on our detective hats and start cracking this case! We're going to go through each equation one by one and see if it makes sense in the context of our tomato patch and vegetable garden. Remember, we're looking for an equation that relates the length of the tomato patch (which we'll likely represent with 'x') to the total garden area of 170 square feet. This process might seem a bit like trial and error, but it's a systematic way to approach the problem. Let's dive in!

Analyzing Option A: $0=3 x^2+2 x+180$

Let's start with Option A: $0=3 x^2+2 x+180$. Now, the first thing that might jump out at you is the constant term, +180. This term represents a fixed area, and it's positive. In our scenario, the total area of the garden is 170 square feet. If this equation were correct, it would imply that there's already an area of 180 square feet before we even consider the tomato patch! That doesn't quite add up, does it? The area should be subtracted to see if it fits the criteria of the garden area being 170 square feet. So, Option A seems unlikely.

Analyzing Option B: $0=3 x^2-160$

Next up, Option B: $0=3 x^2-160$. This equation looks a bit more promising. We have a term with $x^2$, which likely represents the area of the tomato patch, and we're subtracting 160. This could potentially relate to the total garden area. If we rearrange this equation, we get $3x^2 = 160$. This suggests that $3x^2$ might be the area of the tomato patch and that area is related to the garden's total area. This equation is still a contender, but we need to keep the garden area in mind, and see if this fits the 170 square feet criteria.

Analyzing Option C: $0=3 x^2+10 x+180$

Now, let's consider Option C: $0=3 x^2+10 x+180$. Just like Option A, this equation has a large positive constant term (+180). Again, this suggests an initial area that's already greater than the total garden area, which doesn't fit our problem. Plus, the presence of a '+10x' term doesn't immediately make sense in the context of areas. So, Option C also seems unlikely to be the correct answer.

Analyzing Option D: $0=3 x^2+17 x-160$

Finally, let's examine Option D: $0=3 x^2+17 x-160$. This equation is interesting because it has a negative constant term (-160). This could potentially represent subtracting an area from a larger area, which aligns with our problem. We're trying to find the length of the tomato patch within the 170 square foot garden, so subtracting a value makes logical sense. This equation is a strong candidate and fits the scenario of the problem accurately.

The Final Verdict and Explanation

Okay, guys, after carefully analyzing all the options, we've narrowed it down to the most likely answer. Remember, we were looking for an equation that relates the length of the tomato patch to the total garden area of 170 square feet. We went through each option, considering how the terms in the equation might represent areas and dimensions.

Based on our analysis, Option D: $0=3 x^2+17 x-160$ is the equation that best fits the problem's description. Here's why:

• The $3x^2$ term likely represents the area of the tomato patch, where 'x' is related to its length.
• The +17x term could represent an additional area or adjustment related to the dimensions of the patch.
• Crucially, the -160 term represents subtracting an area. This is essential because it allows us to relate the tomato patch area to the overall garden area of 170 square feet. If we rearrange the equation slightly, we get $3x^2 + 17x = 160$. If you add 10 to both sides, it becomes $3x^2 + 17x + 10 = 170$ which is the overall garden area

Options A and C had large positive constant terms, which didn't make sense in the context of our problem. Option B was closer, but it didn't have the additional '+17x' term, which is likely necessary to fully describe the relationship between the dimensions and areas.

Therefore, the final answer is D. $0=3 x^2+17 x-160$. This equation allows us to find the length of the tomato patch given the total area of the garden, making it the correct choice. Great job, everyone! We solved it!