Cary Calculated The Surface Area Of A Box In The Shape Of A Rectangular Prism. She Wrote The Equation \[$148 - 2(6w + 6h + Hw)\$\] To Represent The Width And Height Of The Box. She Solved For \[$w\$\] And Got \[$w = \frac{74 - 6h}{h

Introduction

In mathematics, calculating the surface area of a three-dimensional object is a fundamental concept that has numerous real-world applications. A rectangular prism, also known as a box, is a common shape used in various fields such as architecture, engineering, and design. In this article, we will delve into the mathematical concept of calculating the surface area of a rectangular prism, focusing on the equation provided by Cary, a math enthusiast. We will explore the equation, solve for the width, and discuss the implications of this mathematical concept.

The Equation for Surface Area

Cary's equation for the surface area of a rectangular prism is given by:

$$148 - 2(6w + 6h + hw)$$

This equation represents the surface area of the box, where $w$ is the width and $h$ is the height. The equation is a combination of the areas of the six faces of the rectangular prism, which are the top and bottom faces, the front and back faces, and the left and right faces.

Solving for Width

To solve for the width, we need to isolate the variable $w$ in the equation. We can start by simplifying the equation:

$$148 - 2(6w + 6h + hw)$$

$$148 - 12w - 12h - 2hw$$

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148 -<br/> **Calculating the Surface Area of a Rectangular Prism: A Mathematical Exploration** =========================================================== **Q&A: Calculating the Surface Area of a Rectangular Prism** --------------------------------------------------- **Q: What is the surface area of a rectangular prism?** ------------------------------------------------ A: The surface area of a rectangular prism is the total area of all six faces of the prism. It can be calculated using the formula: 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height. **Q: How do I calculate the surface area of a rectangular prism using the equation provided by Cary?** --------------------------------------------------------------------------------------------- A: To calculate the surface area of a rectangular prism using the equation provided by Cary, you need to substitute the values of l, w, and h into the equation: 148 - 2(6w + 6h + hw). Then, simplify the equation and solve for the surface area. **Q: What is the equation for the surface area of a rectangular prism?** ---------------------------------------------------------------- A: The equation for the surface area of a rectangular prism is: 148 - 2(6w + 6h + hw). **Q: How do I solve for the width (w) in the equation?** --------------------------------------------------- A: To solve for the width (w) in the equation, you need to isolate the variable w. You can start by simplifying the equation: 148 - 2(6w + 6h + hw). Then, combine like terms and solve for w. **Q: What is the formula for the surface area of a rectangular prism?** ---------------------------------------------------------------- A: The formula for the surface area of a rectangular prism is: 2lw + 2lh + 2wh. **Q: How do I calculate the surface area of a rectangular prism using the formula?** --------------------------------------------------------------------------- A: To calculate the surface area of a rectangular prism using the formula, you need to substitute the values of l, w, and h into the formula: 2lw + 2lh + 2wh. Then, simplify the equation and solve for the surface area. **Q: What is the relationship between the surface area and the dimensions of a rectangular prism?** ----------------------------------------------------------------------------------------- A: The surface area of a rectangular prism is directly proportional to the dimensions of the prism. As the dimensions of the prism increase, the surface area also increases. **Q: How do I use the surface area formula to find the dimensions of a rectangular prism?** -------------------------------------------------------------------------------------- A: To use the surface area formula to find the dimensions of a rectangular prism, you need to rearrange the formula to isolate one of the variables. For example, if you know the surface area and the height, you can use the formula to find the length and width. **Q: What are some real-world applications of calculating the surface area of a rectangular prism?** ----------------------------------------------------------------------------------------- A: Calculating the surface area of a rectangular prism has numerous real-world applications, such as: * Architecture: Calculating the surface area of a building to determine the amount of materials needed for construction. * Engineering: Calculating the surface area of a machine or a device to determine its efficiency and performance. * Design: Calculating the surface area of a product to determine its packaging and shipping requirements. **Conclusion** ---------- Calculating the surface area of a rectangular prism is a fundamental concept in mathematics that has numerous real-world applications. By understanding the equation and formula for the surface area, you can solve problems and make informed decisions in various fields. Remember to always simplify the equation and solve for the variable to find the surface area of a rectangular prism.