Solving For Width: Cary's Rectangular Prism Equation
Cary's Rectangular Prism: Unraveling Surface Area and Width
Hey guys, let's dive into a cool math problem! We're going to explore how Cary tackled calculating the surface area of a box, which is a rectangular prism. This kind of stuff is super practical, like when you're figuring out how much wrapping paper you need for a gift! Cary came up with a neat equation, and we're going to break it down step by step. We'll see how she found the width of the box using some clever algebra. So grab your calculators, and let's get started!
Understanding the Problem: Surface Area and Rectangular Prisms
Alright, first things first: what exactly is a rectangular prism? Think of it like a box – it has a length, a width, and a height. The surface area is the total area of all the faces of the box added together. Cary's equation, 148 = 2(6w + 6h + hw), is a mathematical way of representing the surface area of this specific box. The "148" represents the total surface area. The 'w' stands for the width, and the 'h' is the height of the box. The equation is basically saying: "If you add up the areas of all the sides of the box, you get 148". The number 6 is likely related to a fixed length value of the box. The formula 2(6w + 6h + hw) is used to calculate the surface area of the box. The formula can be broken down into the following components: 2(lw + lh + wh), where: l = length, w = width, h = height. In this case, the fixed length = 6. The equation shows us how all the dimensions of the box contribute to the overall surface area. It's like a puzzle where each piece (the dimensions) fits together to create the bigger picture (the surface area).
To truly grasp this, imagine unfolding the box. You'd see six rectangles: the top, bottom, front, back, and two sides. Cary's equation is a smart way of calculating the area of each of these rectangles and then adding them all up. The '2' in the equation comes from the fact that there are two of each identical face (top and bottom, front and back, left and right). So, we're effectively doubling the area of the different face pairs. Understanding the components in a formula is key to figuring out the math problems. The key is knowing how each component contributes to the overall equation. When we have this knowledge, it makes it easier to work out the problem and solve it.
Breaking Down Cary's Equation: Solving for Width
Now, let's get into the nitty-gritty of how Cary solved for the width, 'w'. She cleverly rearranged the equation to isolate 'w'. You know, like detectives in math, we need to unravel the mystery! Cary's result was w = (74 - 6h) / (h + 6). This new equation tells us that to find the width, we need to know the height ('h'). The value 74 is half of the surface area 148. It seems like the length 6 is already included in the equation. The height 'h' is used to solve the width 'w'. This is super useful because it means if we know the height of the box, we can easily calculate its width. Now, let's quickly review how Cary might have arrived at this result. First, she would likely have divided both sides of the original equation 148 = 2(6w + 6h + hw) by 2, resulting in 74 = 6w + 6h + hw. Next, we must isolate terms with 'w'. The equation would be rewritten as 74 - 6h = 6w + hw. This step allows us to get all the 'w' terms on one side and the rest of the terms on the other side. Now, factoring out 'w', we get 74 - 6h = w(6 + h). Finally, to solve for 'w', divide both sides by (6 + h) to get the final answer w = (74 - 6h) / (h + 6). By making these rearrangements, the original equation is rewritten into the new form, which will help us understand the width of the rectangular prism.
Putting It All Together: Practical Applications
So, why is this kind of math important? Well, it's more than just a math problem. Understanding surface area and being able to solve for dimensions has real-world applications. Consider these scenarios: packing boxes for moving, designing packaging for products, or even estimating the amount of paint needed to cover a wall. Knowing how to calculate the surface area lets you figure out how much of a material you'll need. This saves both money and resources. Similarly, the skill of solving for a specific variable, like Cary did with 'w', is valuable in many fields. Whether you're a designer, an engineer, or an architect, this skill is useful. For example, imagine designing a custom-sized box. You might know the desired surface area and the height and want to figure out the width. Cary's method provides a formula for a quick answer. It also teaches us about the power of rearranging equations to reveal useful information. It's like having a mathematical key that unlocks the secrets of different shapes. Remember, math is not just about equations and formulas; it is about problem-solving. By understanding the concepts and techniques, we can be successful in different fields.
Further Exploration: More Complex Shapes and Equations
If you're enjoying this, let's consider some more complex shapes and equations. You can expand your knowledge by experimenting with different geometric shapes. Think about how you would calculate the surface area of a cylinder (like a can) or a cone (like an ice cream cone). The surface area calculation for a cylinder involves the area of the two circular ends and the curved side. The formula is: 2πr² + 2πrh, where 'r' is the radius and 'h' is the height. For a cone, the surface area calculation uses the radius (r), the slant height (s), and the formula is πr(s + r). Similarly, the ability to solve for a variable is a skill that extends to many areas of math. Learn how to manipulate formulas to isolate a specific variable. Practice solving quadratic equations, or even systems of equations, to develop your problem-solving skills. The more you practice, the more comfortable you'll become with algebraic manipulations. It's like building muscles for your brain, guys. The more you practice, the better you get! Remember, math is a journey, not a destination. Keep exploring, keep asking questions, and have fun! You got this!
