Problem 19 - Algebra 2 and Trigonometry

Question

A carton is completely filled with boxes that are 1 foot cubes. The length of the carton is 2 feet greater than the width and the height of the carton is 3 feet. If the carton holds at most 72 cubes, what are the possible dimensions of the carton?

Step-by-Step Solution

Answer

The possible dimensions are (1, 3, 3), (2, 4, 3), (3, 5, 3), and (4, 6, 3).

Step 1: Understand the Problem

We need to find the dimensions (length, width, and height) of a carton that can hold at most 72 cubes that are each 1 cubic foot. The length of the carton is 2 feet greater than its width, and its height is given as 3 feet.

Step 2: Set Up the Variables

Let's denote the width of the carton as \( w \), the length as \( l \), and the height as \( h = 3 \). Then, the length \( l \) can be expressed as \( l = w + 2 \).

Step 3: Write the Volume Inequality

The volume of the carton is the product of its length, width, and height, which is given as \( V = l \times w \times h \). We know the volume cannot exceed 72 cubic feet, so we have:\[ (w + 2) \times w \times 3 \leq 72 \]

Step 4: Simplify the Inequality

Simplify the inequality:\[ 3(w + 2)w \leq 72 \]Divide both sides by 3:\[ (w + 2)w \leq 24 \]

Step 5: Solve the Quadratic Inequality

Expand the inequality:\[ w^2 + 2w \leq 24 \]Rearrange into standard form:\[ w^2 + 2w - 24 \leq 0 \]Solve this by factoring or using the quadratic formula. The quadratic factors to:\[ (w - 4)(w + 6) \leq 0 \]

Step 6: Determine the Range of Width \( w \)

The solutions to the quadratic inequality \( (w - 4)(w + 6) \leq 0 \) occur between the roots of the equation, meaning \(-6 \leq w \leq 4 \). Since the width must be positive, reduce this to \( 0 \leq w \leq 4 \).

Step 7: Calculate Corresponding Lengths

For possible integer widths between 1 and 4 (inclusion of 0 makes no sense because the width cannot be zero), the lengths are computed as \( l = w + 2 \). Therefore:- If \( w = 1 \), \( l = 3 \)- If \( w = 2 \), \( l = 4 \)- If \( w = 3 \), \( l = 5 \)- If \( w = 4 \), \( l = 6 \)

Step 8: Verify Each Dimension Set

Check each set of dimensions:1. \( w = 1, l = 3, h = 3 \) gives \( 1 \times 3 \times 3 = 9 \) cubes2. \( w = 2, l = 4, h = 3 \) gives \( 2 \times 4 \times 3 = 24 \) cubes 3. \( w = 3, l = 5, h = 3 \) gives \( 3 \times 5 \times 3 = 45 \) cubes4. \( w = 4, l = 6, h = 3 \) gives \( 4 \times 6 \times 3 = 72 \) cubesAll these dimension pairs are valid as they do not exceed 72 cubes.

Key Concepts

Volume of a Rectangular Prism

Understanding the volume of a rectangular prism is key when working with three-dimensional objects like a carton. A rectangular prism is essentially a box, and its volume tells us how much space it contains inside. The formula for calculating the volume of a rectangular prism is straightforward: it is the product of its length, width, and height. To express this mathematically:

\[ V = l \times w \times h \]
Where:

• **\( l \)** is the length
• **\( w \)** is the width
• **\( h \)** is the height

In the context of our problem, we have a specified height of 3 feet, while the length is 2 feet more than the width. The maximum volume the carton can hold is 72 cubic feet, guiding us to solve the inequality that represents this volume limit. By understanding these relationships, we can better visualize how changing the width or length would affect the overall size of the carton.

Solving Quadratic Equations

Whenever you have expressions involving squares, you're often dealing with quadratic equations. These equations take the form \( ax^2 + bx + c = 0 \). In problems like this, they help us figure out dimensions without going overboard in volume. Quadratic equations can be tackled using different methods:

• **Factoring**
• **Quadratic formula**
• **Completing the square**

For our exercise, once we simplified the volume inequality, we ended up with a quadratic equation: \( w^2 + 2w - 24 \leq 0 \). This kind of equation can be solved by finding values for \( w \) that satisfy the condition. Solving quadratics means pinpointing those moments when our factorized terms, like \((w - 4)(w + 6) \leq 0\), squeeze into a range that doesn't encroach on the maximum volume capacity. By setting the factors to zero, you determine where the transitions occur and thereby outline safe spots for your dimensions.

Factoring Quadratics

Factoring quadratics is a method used to simplify and solve quadratic equations. It involves rewriting a quadratic equation into a product of two binomials. This is particularly useful when you want to find solutions or roots of the equation. In our task of determining the dimensions of the carton, factoring allows us to express the quadratic inequality as:

\[ (w - 4)(w + 6) \leq 0 \]
By setting each factor equal to zero, you find potential solutions or critical points:

• \( w - 4 = 0 \Longrightarrow w = 4 \)
• \( w + 6 = 0 \Longrightarrow w = -6 \)

However, since a width cannot be negative, we only consider positive results, telling us that the width should be between 0 and 4 feet. Factoring provides a neat route to finding acceptable dimensions without needing extraneous methods. It directly offers insights into feasible ranges and makes working with quadratics a lot more manageable. It's not just arithmetic; it's the tool that crafts practicality out of algebra.