Rectangle Dimensions: Perimeter 80x, Base 2x Height

Hey guys! Let's dive into a classic geometry problem that's sure to get your brains buzzing. We've got a rectangle, and we know its perimeter is a neat 80x. On top of that, we're told a crucial piece of info: the base of this rectangle is exactly twice its height. Our mission, should we choose to accept it (and we totally should!), is to figure out what those dimensions – the height and the base – are, all expressed in terms of that mysterious variable x. This isn't just about crunching numbers; it's about understanding how different parts of a geometric shape relate to each other, especially when a perimeter is involved. So, grab your notebooks, maybe a trusty calculator, and let's break down how to solve this puzzle step-by-step.

To start, let's lay down the fundamental rules of rectangles and their perimeters. You guys know that a rectangle has four sides, with opposite sides being equal in length. The height (or width) and the base (or length) are the two distinct dimensions we're interested in. The perimeter is simply the total distance around the outside of the shape. Mathematically, the formula for the perimeter (P) of a rectangle is P = 2 * (base + height). We've been given that our rectangle's perimeter is P = 80x. So, we can plug this into our formula: 80x = 2 * (base + height). This equation is our starting point, our anchor in this problem. It connects the given perimeter to the dimensions we need to find. Now, remember that condition: the base is twice the height. We can express this relationship with a simple equation: base = 2 * height. This second equation is key because it allows us to substitute one variable for another, simplifying our first equation significantly. We're essentially creating a system of equations, a common technique in algebra, to solve for unknown values. By combining the general perimeter formula with the specific relationship between the base and height, we can isolate and solve for the dimensions. Think of it like this: we have two clues about our rectangle, and by using both, we can uniquely identify its size. So, let's get ready to use these tools to find those dimensions in terms of x.

Now, let's get down to business and actually solve for those dimensions. We have our two core equations: 80x = 2 * (base + height) and base = 2 * height. The easiest way to proceed is by substituting the second equation into the first. Since we know that 'base' is the same thing as '2 * height', we can replace 'base' in the perimeter equation with '2 * height'. Let's do that: 80x = 2 * ((2 * height) + height). See what we did there? We swapped 'base' out for its equivalent expression in terms of 'height'. This is a powerful move because it means our perimeter equation now only has one variable: 'height'. Let's simplify the expression inside the parentheses first: (2 * height) + height = 3 * height. So, the equation becomes: 80x = 2 * (3 * height). Now, multiply the 2 by the 3: 80x = 6 * height. Bingo! We've isolated the term with 'height'. To find the value of 'height' itself, we just need to divide both sides of the equation by 6. So, height = (80x) / 6. We can simplify this fraction. Both 80 and 6 are divisible by 2: height = (40x) / 3. And there you have it – the height of the rectangle, expressed in terms of x! It's 40x/3. Now that we've found the height, finding the base is a piece of cake using our relationship base = 2 * height. Simply substitute the value of height we just found: base = 2 * (40x / 3). Multiplying this out, we get base = 80x / 3. So, the dimensions of our rectangle are a height of 40x/3 and a base of 80x/3. It's pretty cool how we can use a couple of algebraic equations to nail down these geometric properties, right?

Let's take a moment to check our work and make sure these dimensions actually give us the perimeter we started with. We found that the height is 40x/3 and the base is 80x/3. Remember our perimeter formula: P = 2 * (base + height). Let's plug in our calculated dimensions: P = 2 * ( (80x/3) + (40x/3) ). First, let's add the terms inside the parentheses. Since they already have a common denominator (which is 3), we can just add the numerators: 80x + 40x = 120x. So, the expression inside becomes 120x/3. Now, we can simplify 120x/3 by dividing 120 by 3, which gives us 40x. So, the equation now is: P = 2 * (40x). Finally, multiply by 2: P = 80x. And guess what? That's exactly the perimeter we were given at the start! 80x! This confirms that our calculated dimensions for the height (40x/3) and the base (80x/3) are absolutely correct. It's always a good idea to double-check, especially in math problems. It builds confidence and helps catch any silly mistakes. So, to recap, for a rectangle with a perimeter of 80x where the base is twice the height, the height is 40x/3 and the base is 80x/3. Pretty neat, huh?

Now, let's look at the multiple-choice options provided and see which one matches our findings. Our calculated dimensions are: height = 40x/3 and base = 80x/3. Let's convert these to decimals to compare them easily with the options, keeping in mind that 40/3 is approximately 13.33 and 80/3 is approximately 26.67.

A. $h=10 x ; b=10 x$. If the height and base were both $10x$, the base wouldn't be twice the height. Plus, the perimeter would be $2(10x + 10x) = 2(20x) = 40x$, which isn't $80x$.

B. $h=13.3 x ; b=26.6 x$. Let's see if the base is twice the height here: $2 * 13.3x = 26.6x$. Yes, it is! Now let's check the perimeter: $P = 2 * (26.6x + 13.3x) = 2 * (39.9x)$. This is very close to $2 * (40x) = 80x$. If we use the fractions, $h = 40x/3$ and $b = 80x/3$, then $b = 2h$ is true, and $P = 2(80x/3 + 40x/3) = 2(120x/3) = 2(40x) = 80x$. The decimal approximations in option B are very close to our fractional answers, making it the likely correct choice. The slight difference is due to rounding $40/3$ and $80/3$ to one decimal place.

C. $h=10.3 x , b=20.6 x$. Here, the base is twice the height ($2 * 10.3x = 20.6x$). Let's check the perimeter: $P = 2 * (20.6x + 10.3x) = 2 * (30.9x) = 61.8x$. This is not $80x$.

D. $h=5 x ; b=25 x$. The base is not twice the height ($2 * 5x = 10x$, not $25x$). The perimeter would be $2(25x + 5x) = 2(30x) = 60x$, which is not $80x$.

Comparing our calculated exact values ($h=40x/3, b=80x/3$) with the options, option B provides the dimensions that satisfy both conditions (base is twice the height, and the perimeter is $80x$, using the fractional equivalents $h = 40x/3 ext{ and } b = 80x/3$). The decimal values in option B are approximations of these fractions.

So, the correct dimensions for the rectangle, in terms of x, are a height of approximately 13.3x and a base of approximately 26.6x. This corresponds to option B. Great job working through this, guys! Remember, breaking down problems into smaller, manageable steps and using the right formulas is key to conquering any math challenge.