Problem 30 - Algebra 2

Question

The length of the cargo space in a sport-utility vehicle is 4 inches greater than the height of the space. The width is sixteen inches less than twice the height. The cargo space has a total volume of 55,296 cubic inches. Use a rectangular prism to model the cargo space. Write a polynomial function that represents the volume of the cargo space.

Step-by-Step Solution

Answer

The polynomial function is \( 2h^3 - 8h^2 - 64h = 55,296 \).

Step 1: Define Variables

Let the height of the cargo space be \( h \) inches. Then, the length of the cargo space can be expressed as \( h + 4 \) inches since it is 4 inches more than the height.

Step 2: Express Width in Terms of Height

Given that the width is sixteen inches less than twice the height, express the width as \( 2h - 16 \) inches.

Step 3: Write the Volume Expression

The volume \( V \) of a rectangular prism is calculated by multiplying the length, width, and height. Thus, the volume of the cargo space can be written as: \[ V = h \times (h + 4) \times (2h - 16) \]

Step 4: Set Up the Polynomial Equation

We know the volume is 55,296 cubic inches. Therefore, set up the equation: \[ h \times (h + 4) \times (2h - 16) = 55,296 \]

Step 5: Simplify the Polynomial

First, simplify the polynomial expression by expanding it. Start by expanding \( (h + 4)(2h - 16) \): \[ h(h(2h - 16) + 4(2h - 16)) = h(2h^2 - 16h + 8h - 64) = h(2h^2 - 8h - 64) \]. Then multiply by \( h \): \[ 2h^3 - 8h^2 - 64h \].

Step 6: Write the Final Polynomial Equation

The polynomial equation representing the volume is \( 2h^3 - 8h^2 - 64h = 55,296 \). This equation can be solved for the height \( h \) to check its suitability.

Key Concepts

Rectangular Prism Volume

The concept of volume for a rectangular prism is a fundamental aspect of geometry and is frequently used in real-world situations. A rectangular prism is simply a three-dimensional shape with six faces that are all rectangles. The volume of a rectangular prism is the amount of space inside the shape and is calculated by multiplying its three dimensions: length, width, and height.

To visualize, imagine a box. The length is the longer side of the base, the width is the shorter side of the base, and the height is the side that extends upwards. This multiplication gives \[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \]

In our exercise, the dimensions aren't given directly but are instead expressed in relation to a common variable, the height. By understanding how each measurement relates to the height, we can model the cargo space as a polynomial equation which represents the volume.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. They are a vital part of algebra that allow you to solve for unknown variables. In the context of our problem, an algebraic equation is used to express the volume of the rectangular prism.

By transforming the relationship between the dimensions of the prism into expressions with the variable height \( h \), we can write an equation for volume. The exercise provides external connections between the dimensions:

• Length is expressed as \( h + 4 \)
• Width is \( 2h - 16 \)
• Height is \( h \)

These relationships are combined to form the polynomial equation \[ h(h + 4)(2h - 16) = 55,296 \]

This equation encapsulates the volume of the cargo space, using algebra to enable us to solve for \( h \), the height of the cargo space. Solving this equation requires simplifying and rearranging, a process that aids in finding the dimension responsible for the given volume.

Variable Expressions

Variable expressions are algebraic expressions that include numbers, variables, and operation symbols. They are essential in expressing relationships in terms that can be manipulated mathematically.

In our exercise, variable expressions are used to express the dimensions of the cargo space. Starting with one unknown, the height \( h \), expressions are formed for each dimension of the rectangular prism relative to \( h \).

• The height remains \( h \)
• The length is \( h + 4 \), indicating a direct relationship
• The width is \( 2h - 16 \), showcasing a more complex dependency on height

These variable expressions allow us to set up the equation for the volume and, by extension, solve for \( h \).

Manipulating these expressions helps us understand and explore the properties of the rectangular prism in terms of height. Grasping this interaction is crucial in translating real-world scenarios into mathematical models that can then be solved using algebra.