Problem 44 - Algebra 2

Question

Antonio is preparing to make an ice sculpture. He has a block of ice that he wants to reduce in size by shaving off the same amount from the length, width, and height. He wants to reduce the volume of the ice block to 24 cubic feet. Write a polynomial equation to model this situation.

Step-by-Step Solution

Answer

Polynomial equation for the volume: \\( (l-x)(w-x)(h-x) = 24 \\)

Step 1: Understand the Problem

We need to model the volume reduction of a block. We start with a block of ice and reduce its dimensions by the same amount from length, width, and height.

Step 2: Define the Variables

Let the original dimensions of the block be length \(l\), width \(w\), and height \(h\). Let \(x\) be the amount to shave off each dimension.

Step 3: Write the Expression for New Dimensions

After reducing each dimension by \(x\), the new dimensions become \(l-x\), \(w-x\), and \(h-x\).

Step 4: Write the Volume Equation

The volume of the new block is given by \((l-x)(w-x)(h-x) = 24\), where 24 cubic feet is the target volume.

Step 5: Formulate the Polynomial Equation

Expand the expression \((l-x)(w-x)(h-x) = 24\) to form a polynomial. This represents the equation to model the sculpture volume reduction.

Key Concepts

Volume Reduction

Volume reduction in this context refers to decreasing the size of a three-dimensional object, such as a block of ice, by reducing one or more of its dimensions. It’s a common technique used in practical applications like sculpture making. To achieve a target volume when reducing an object:

• Determine how much to reduce each dimension equally to achieve the desired smaller volume.
• In Antonio's problem, he's shaving the same amount from length, width, and height.
• The original volume is reworked down to 24 cubic feet once the dimensions are reduced smoothly.

The goal is to perform these dimensional reductions while keeping the shape proportional and achieving the specific volume target.

Variable Definition

In mathematical modeling, defining variables is crucial as it creates a framework for solving equations. In Antonio's case:

• The original dimensions of the ice block are represented by variables: length ( l ), width ( w ), and height ( h ).
• A new variable ( x ) is introduced, representing the uniform length to be reduced from each dimension to reach the target volume.
• These variables allow for constructing an equation that expresses the relationship between dimensional changes and volume.

This systematic approach ensures clarity, allowing one to combine and manipulate these values to form the required polynomial equation model.

Dimension Reduction

Dimension reduction involves altering the size of the object by uniformly trimming its measures such as length, width, and height. For Antonio's ice block:

• The length, width, and height of the block are each reduced by x units, leading to new dimensions of (l-x) , (w-x) , and (h-x) respectively.
• This operation is essential for achieving a smaller, more manageable object volume.
• Accurate reduction helps maintain the block's proportions and is key to achieving the targeted reduction in volume.

Dimension reduction is a straightforward yet powerful technique used in various applications beyond sculptures, including storage and manufacturing adjustments.

Cubic Measurement

Cubic measurement is a process of determining the volume of a three-dimensional object, expressed in cubic units. For defined shapes such as Antonio’s ice block:

• The original volume is calculated using the dimensions as l times w times h .
• After reducing each dimension by x , the new volume becomes (l-x)(w-x)(h-x) = 24 cubic feet.
• This equation now needs solving to find x , which decides how much each dimension is reduced by while achieving the precise 24 cubic feet.

Cubic measurement forms the backbone of understanding and working with volumes in architectural and design contexts.