Understanding the Surface Area of a Box: A Dive into Geometry

Have you ever looked at a box and wondered how to calculate its surface area? For students preparing for the Officer Aptitude Rating (OAR), mastering these concepts can pave the way to success. Let’s roll up our sleeves and dive into the world of geometry, shall we? Today, we’ll focus on a specific problem that involves a closed rectangular box with a square base.

Breaking Down the Problem

Imagine you have a closed rectangular box with a square base. It has a height of 3 inches and a volume of 48 cubic inches. The inquiry here is straightforward yet fundamental: What is the surface area of this box? To solve this, we need to carefully analyze the data provided.

Let’s denote the side length of the square base as "s." Since the box has a square base, both the length and width will be equal, making our calculations a bit simpler. The height is given—3 inches—so we can represent the volume formula of a rectangular prism, which tells us that:

Volume = length × width × height.

Since our box’s base is square, we rewrite this as:

Volume = s^2 × height = s^2 × 3.

Given that the volume is 48 cubic inches, we can set up the equation:

s^2 × 3 = 48.

Next, let’s isolate s² by dividing both sides by 3. Easy peasy, right?

s² = 48 / 3 = 16.

Finding the Side Length

Now, here’s where it gets a bit exciting—let’s find the length of one side of our square base. To do this, we take the square root of 16:

s = 4 inches.

With the side length in hand, we can now calculate the box's surface area. But before we get into the math, let’s reflect: why is knowing these calculations important? For many students, geometry can be daunting, yet it holds practical applications in real life, from designing structures to packaging products.

Calculating Surface Area

The surface area of a rectangular box uses this formula:

Surface Area = 2(length × width + length × height + width × height).

Knowing our box has a square base means length and width both equal s. So, let’s plug our values into the surface area formula:

• Length × Width = s × s = 4 × 4 = 16 square inches
• Length × Height = s × height = 4 × 3 = 12 square inches
• Width × Height = s × height = 4 × 3 = 12 square inches

Now, putting it all together:

Surface Area = 2(16 + 12 + 12), which simplifies to:

Surface Area = 2(40) = 80 square inches.

Final Thoughts

And that’s it! The box's surface area is 80 square inches, which matches one of our answer choices. How cool is that?

Understanding these calculations will not only help tackle OAR practice questions effectively but will also elucidate a fundamental concept in geometry. So, whether you're preparing for an exam or simply want to sharpen your math skills, mastering how to determine the surface area of a box is definitely worth your time.

Remember, geometry isn't just about getting the right calculations; it’s about understanding the world around us and solving real-life problems. Now, take that knowledge and run with it—because you've got this!