Problem 43 - College Algebra with Modeling and Visualization

Question

Find the volume and surface area of a rectangular box with length \(L\), width \(W\), and height \(H\). \(L=x, W=2 y, H=3 z\)

Step-by-Step Solution

Answer

Volume: \(6xyz\), Surface Area: \(4xy + 6xz + 12yz\).

Step 1: Understanding the Problem

To find the volume and surface area of a rectangular box, we need to use the formulas for both volume and surface area. The given dimensions for the box are replaced by algebraic expressions: length \( L = x \), width \( W = 2y \), and height \( H = 3z \).

Step 2: Calculating the Volume

The formula for the volume \( V \) of a rectangular box is \( V = L \times W \times H \). Substitute the given expressions into the formula: \[ V = x \times 2y \times 3z = 6xyz \]

Step 3: Calculating the Surface Area

The formula for the surface area \( A \) of the box is \[ A = 2(LW + LH + WH) \]Substitute the given expressions into the formula: \[ A = 2(x \cdot 2y + x \cdot 3z + 2y \cdot 3z) = 2(2xy + 3xz + 6yz) \]Simplify: \[ A = 4xy + 6xz + 12yz \]

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that use numbers, variables, and operation symbols. In our problem, the dimensions of the rectangular box are given as algebraic expressions:

• Length: \(L = x\)
• Width: \(W = 2y\)
• Height: \(H = 3z\)

These expressions allow us to represent the dimensions with variables, providing flexibility to calculate volume and surface area for any values of \(x\), \(y\), and \(z\). Using algebraic expressions is beneficial when we have unknown or changing dimensions, making this a powerful tool in many mathematical problems.

Geometric Formulas

Geometric formulas are essential for solving problems related to shapes and their properties. They allow us to calculate measurements like area, volume, and surface area. For a rectangular box, the key formulas are:

• Volume \(V\): \(V = L \times W \times H\)
• Surface Area \(A\): \(A = 2(LW + LH + WH)\)

In our exercise, we employ these formulas to determine the volume and surface area using algebraic expressions. After substituting the dimensions, we have:

• Volume: \(V = x \times 2y \times 3z = 6xyz\)
• Surface Area: \(A = 2(2xy + 3xz + 6yz) = 4xy + 6xz + 12yz\)

Understanding and applying these geometric formulas is crucial in finding the desired measurements of the box.

Problem Solving

Problem solving involves identifying and applying the appropriate methods and strategies to find a solution. In this exercise, the problem-solving process includes:

• Interpreting the problem: Understand that we need to find both volume and surface area of the box.
• Using the right formulas: Employ the geometric formulas specific to a rectangular box.
• Substituting and simplifying: Replace dimensions with the given algebraic expressions and perform operations step by step.

An effective problem-solving strategy involves breaking the problem into manageable parts, making it easier to handle complex calculations. By using algebraic expressions and formulas efficiently, we can tackle similar geometric problems with confidence.