Find The Domain: Rectangle Area Function

Hey guys! Today, we're diving into a super cool math problem involving rectangles and functions. We've got a scenario where the area of a rectangle is fixed at 80 square units. The relationship between its height ($H$) and width ($w$) is given by the function:

$H(w) = \frac{80}{w}$

This function tells us that no matter what the width is, we can always find the corresponding height to maintain that 80 square unit area. It's like a puzzle where if you know one piece (the width), you can figure out the other (the height).

Now, the big question is: What is the domain of this function? Let's break it down.

Understanding Domain in Functions

Before we jump into solving, let's get clear on what we mean by the 'domain' of a function. In simple terms, the domain is the set of all possible input values for a function. Think of it as the allowed values for the variable on the x-axis (or in our case, the 'w' axis). For a function to be mathematically valid and make sense in a real-world context, its inputs usually have some restrictions. We can't just plug in any number and expect a meaningful result, especially when we're dealing with physical dimensions like width.

In mathematics, we often look out for a few common culprits that can restrict a domain. These include:

• Division by zero: You can never divide by zero. If a function has a variable in the denominator, we must ensure that the denominator never equals zero. This would make the function undefined.
• Square roots of negative numbers: In the realm of real numbers, we can't take the square root of a negative number. So, if a function involves a square root, the expression inside the square root must be greater than or equal to zero.
• Logarithms of non-positive numbers: Logarithms are only defined for positive numbers.

For our specific function, $H(w) = \frac{80}{w}$, the main thing we need to worry about is that pesky division by zero. The variable $w$ is in the denominator.

Analyzing the Function $H(w) = \frac{80}{w}$

Our function is $H(w) = \frac{80}{w}$. Here, $w$ represents the width of the rectangle. Let's think about what kind of values $w$ can take.

First, the most crucial mathematical restriction comes from the denominator. The width, $w$, cannot be zero. If $w$ were 0, we'd be dividing 80 by 0, which is an undefined operation. So, $w \neq 0$ is our first rule.

Now, let's bring in the real-world context. We're talking about the width of a rectangle. Can a rectangle have a negative width? Not really, guys. In geometry and in practical terms, dimensions like length, width, and height are always considered to be positive values. A width of 0 means the rectangle has collapsed into a line, and a negative width doesn't correspond to any physical measurement we'd encounter. Therefore, the width $w$ must be greater than zero.

So, we have two conditions that seem important: $w \neq 0$ and $w > 0$. When we combine these, the condition $w > 0$ already excludes $w=0$. This means that the only restriction we need to consider from a practical standpoint is that the width must be a positive number.

Putting it All Together: The Domain

The domain of the function $H(w) = \frac{80}{w}$ is the set of all possible values for $w$ that make sense both mathematically and in the context of a rectangle's dimensions. Based on our analysis:

1. Mathematical Restriction: $w \neq 0$ (to avoid division by zero).
2. Real-world Context: $w > 0$ (width must be a positive dimension).

Combining these, the most restrictive and relevant condition is $w > 0$. This means the width can be any positive number, no matter how small (but not zero) or how large.

Let's look at the options provided:

A. $w > 0$ B. $w > 1$ C. $w < 1$ D. $w \leqslant 0$

Option A, $w > 0$, perfectly captures our findings. It ensures that the width is a positive value, making the function mathematically defined and physically meaningful for a rectangle.

Option B ($w > 1$) is too restrictive. A rectangle can have a width between 0 and 1 (e.g., a width of 0.5 units), and the function would still be valid.

Option C ($w < 1$) is also too restrictive and incorrect because it includes values between 0 and 1 but excludes all values greater than or equal to 1, which are perfectly valid widths.

Option D ($w \leqslant 0$) is completely wrong. It includes zero and negative numbers, which are not allowed for the width of a rectangle and would also lead to mathematical issues (division by zero).

Conclusion

So, the domain of the function $H(w) = \frac{80}{w}$, considering the context of a rectangle's width, is $w > 0$. This means the width can be any positive real number. Whether the width is 0.1, 5, 100, or any other positive number, the function will yield a valid height to maintain an area of 80 square units. It's a neat way to see how math functions can represent real-world scenarios, but we always need to keep those real-world constraints in mind! Keep practicing, and you'll be a domain master in no time!

Keywords: domain of a function, function domain, rectangle area, mathematical function, real-world context, width, height, division by zero, positive numbers, inequalities, algebra, functions, math help, problem-solving.

Discussion Category: Mathematics