Problem 13 - Algebra A Combined Function

Question

The surface area of a rectangular box is given by the polynomial \(2 H L+2 L W+2 H W\) and is measured in square units. In business, surface area is often calculated to help determine cost of materials. A rectangular box is to be constructed to hold a new camcorder. The box is to have dimensions 5 inches by 4 inches by 9 inches. Find the surface area of the box.

Step-by-Step Solution

Answer

The surface area of the box is 202 square inches.

Step 1: Identify Given Dimensions

The problem states that the dimensions of the rectangular box are 5 inches (Height), 4 inches (Width), and 9 inches (Length). Let \( H = 5 \), \( W = 4 \), and \( L = 9 \).

Step 2: Recall the Surface Area Formula

The formula to calculate the surface area of a rectangular box is given as \(2HL + 2LW + 2HW\). This formula accounts for the areas of all six faces of the box.

Step 3: Substitute Dimensions into Formula

Substitute the values \( H = 5 \), \( W = 4 \), and \( L = 9 \) into the formula: \[2(5)(9) + 2(9)(4) + 2(5)(4)\].

Step 4: Calculate Each Term in the Formula

Calculate each product separately:- \(2(5)(9) = 90\)- \(2(9)(4) = 72\)- \(2(5)(4) = 40\)

Step 5: Sum the Calculated Areas

Add the calculated term values from Step 4:\(90 + 72 + 40\).

Step 6: Find the Total Surface Area

Calculate the total: \(90 + 72 + 40 = 202\). Therefore, the total surface area of the box is 202 square inches.

Key Concepts

Surface Area Formula

The surface area formula for a rectangular box is a crucial tool in geometry. It helps us calculate the total area covered by the outer layer of the box. This formula is written as \(2HL+2LW+2HW\), where \(H\), \(L\), and \(W\) are the height, length, and width of the box, respectively. Each component of the formula represents the area of different pairs of opposite faces.

• \(2HL\): Represents the area of the front and back faces.
• \(2LW\): Represents the area of the top and bottom faces.
• \(2HW\): Represents the area of the side faces.

When you add these three components together, you get the total surface area of the rectangular box. This formula is fundamental in various applications, from architecture to packaging, as it aids in calculating materials needed or costs.

Rectangular Box

A rectangular box, also known as a rectangular prism, is a three-dimensional shape with six faces, all of which are rectangles. It is defined by three dimensions: length, width, and height. Because its faces are rectangles, each pair of opposite faces is congruent.
This simple geometric shape is found everywhere in the real world. Common examples include:

• Cardboard boxes used for shipping and packaging.
• Bricks in construction.
• Books that we read.

Understanding the properties of a rectangular box is important not just for solving mathematical problems, but also for practical tasks like building objects, determining volumes for storage, and even wrapping gifts.

Polynomial

In mathematics, a polynomial is an expression consisting of variables and coefficients, combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The expression \(2HL + 2LW + 2HW\) is an example of a polynomial. Here, the variables \(H\), \(L\), and \(W\) represent the dimensions of a rectangular box, and the numbers in front of them are coefficients.
Polynomials are used extensively in algebra to describe relationships, solve equations, and model real-world phenomena. They can appear in simple forms, like linear expressions, or more complex forms with higher degrees. With their versatile nature, polynomials are foundational to understanding more complex mathematical concepts, and are especially useful in calculating things like total surface area as we did in this exercise.

• The formula \(2HL + 2LW + 2HW\) simplifies surface area calculations.
• It shows how multiple aspects of a rectangle can be represented in a unified expression.