Rectangle Area Function: L(x)=2x+1, W(x)=x+4

Hey guys! Ever wondered how to find the area of a rectangle when its length and width are described by functions? It's actually pretty straightforward, and today we're going to break it down. So, you've got a rectangle, right? Its length is given by the function $l(x)=2x+1$, and its width is given by $w(x)=x+4$. The big question is, which function defines the area of this rectangle? Let's dive in and figure this out together!

Understanding Rectangle Area

First off, let's get our heads around the basics. The area of a rectangle is fundamental in geometry. You likely learned way back in school that area is calculated by multiplying the length by the width. The formula is simple: $A = l imes w$. Now, in this problem, our length and width aren't just fixed numbers; they're expressed as functions of some variable, $x$. This means the dimensions of our rectangle can change depending on the value of $x$. Our goal is to find a new function, let's call it $a(x)$, that represents the area of this rectangle for any given $x$. So, if $l(x)$ is the length and $w(x)$ is the width, then the area function $a(x)$ will simply be the product of these two functions: $a(x) = l(x) imes w(x)$. It's like plugging in the algebraic expressions for length and width directly into our area formula. This is where the magic of functions comes in – they allow us to represent relationships and calculations that can adapt to different inputs. So, for our specific problem, we need to multiply $(2x+1)$ by $(x+4)$. Don't let the functions scare you; it's just algebraic multiplication. We'll use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last), to multiply these two binomials. This process will give us a new quadratic function that describes the area of our rectangle based on the value of $x$. Remember, the function for area is derived directly from the fundamental formula $A=lw$. So, the first step is always to recognize that you need to perform multiplication of the given expressions for length and width.

Calculating the Area Function

Alright, so we know the area $A$ is found by multiplying the length $l$ and the width $w$. We're given $l(x) = 2x+1$ and $w(x) = x+4$. To find the function that defines the area, $a(x)$, we simply multiply these two functions together: $a(x) = l(x) imes w(x)$.

So, we need to calculate $(2x+1) imes (x+4)$. Let's use the distributive property (or FOIL) to do this:

• First: Multiply the first terms in each binomial: $(2x) imes (x) = 2x^2$
• Outer: Multiply the outer terms: $(2x) imes (4) = 8x$
• Inner: Multiply the inner terms: $(1) imes (x) = x$
• Last: Multiply the last terms: $(1) imes (4) = 4$

Now, we add all these results together: $2x^2 + 8x + x + 4$.

Notice that we have two terms with $x$ in them ($8x$ and $x$). We can combine these like terms: $8x + x = 9x$.

So, the final function for the area is $a(x) = 2x^2 + 9x + 4$.

This function, $a(x)=2x^2+9x+4$, now defines the area of the rectangle for any value of $x$. If you plug in a specific value for $x$, say $x=3$, you can find both the length and width at that point, and then calculate the area. Or, you can directly plug $x=3$ into the area function $a(x)$ and get the same result. Pretty neat, huh? This shows how functions can be combined to represent more complex relationships. We’ve taken two linear functions and, through multiplication, derived a quadratic function representing the area. This is a common pattern when dealing with geometric shapes described by variable dimensions.

Evaluating the Options

Now that we've calculated the area function ourselves, let's look at the options provided:

A. $a(x)=2 x^2+9 x+4$ B. $a(x)=3 x+5$ C. $a(x)=2 x^2+5 x+4$

Comparing our result, $a(x) = 2x^2 + 9x + 4$, with the given options, we can see that Option A matches our calculation perfectly.

Option B, $a(x)=3x+5$, looks like it might come from adding the length and width functions ($l(x)+w(x) = (2x+1) + (x+4) = 3x+5$), which is not how you calculate area.

Option C, $a(x)=2x^2+5x+4$, has the correct $x^2$ term and the constant term, but the middle term ($5x$) is incorrect. This could be a common mistake if someone mixed up the outer and inner products during the FOIL method.

Therefore, the function that correctly defines the area of the rectangle is $a(x)=2 x^2+9 x+4$. This reinforces the importance of careful algebraic manipulation when working with functions. Remember, the fundamental formulas of mathematics are your best friends when tackling these kinds of problems. Stick to the basics, perform your calculations step-by-step, and you'll arrive at the correct answer every time. It's all about understanding the underlying principles and applying them systematically. So, next time you see a problem like this, you'll know exactly what to do!

Conclusion

So there you have it, guys! We've successfully determined the function that defines the area of a rectangle when its length and width are given by functions. The key takeaway is to remember the fundamental formula for the area of a rectangle, $A = l imes w$, and then apply it using the given functions $l(x)=2x+1$ and $w(x)=x+4$. By multiplying these two binomials using the distributive property (FOIL), we arrived at the correct area function: $a(x) = 2x^2 + 9x + 4$. This matches Option A. It's crucial to perform the algebraic multiplication carefully to avoid errors. This exercise highlights how mathematical functions can be combined and manipulated to model real-world concepts, even something as simple as the area of a rectangle. Keep practicing these types of problems, and you'll become a function-whiz in no time! Remember, math is all about understanding the relationships between different quantities, and functions are a powerful tool for describing those relationships. Don't be afraid to break down problems into smaller steps and double-check your work. You got this!