Problem 45 - 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. How much should he take from each dimension?

Step-by-Step Solution

Answer

Antonio should shave off approximately 1 foot from each dimension.

Step 1: Define the Initial Dimensions and Target Volume

Let the initial dimensions of the ice block be given as \( l \, \text{ft} \), \( w \, \text{ft} \), and \( h \, \text{ft} \). The initial volume of the ice block is \( V = l \times w \times h \). Antonio wants to reduce this volume to 24 cubic feet by shaving off \( x \, \text{ft} \) from each dimension.

Step 2: Set Up the Volume Equation After Reduction

The new dimensions of the ice block will be \( l - x \), \( w - x \), and \( h - x \). The equation for the reduced volume is: \((l-x) \cdot (w-x) \cdot (h-x) = 24\).

Step 3: Solve the Equation for Reduction Amount

If \( l = w = h = a \) as one example, then originally \( a^3 \) is the volume of the block. The reduced volume equation becomes \((a-x)^3 = 24\). Solve for \( x \) considering a sample \( a \). For example, if the initial volume was approximately \( a^3 = 64 \, \text{cubic feet}\), then \( a = 4 \). Substitute into \((4-x)^3 = 24\), leading to approximate value \( x = 1\) through trial and error or solving cubic equations.

Key Concepts

Volume Calculation

Volume calculation is a fundamental concept involving the determination of the amount of space a three-dimensional object occupies. When dealing with a block or a rectangular prism, the formula for volume is simple: multiply the length (\( l \)), width (\( w \)), and height (\( h \)). This gives us \( V = l \times w \times h \).

• Length: The longest side of the block.
• Width: The side next to the length, typically shorter.
• Height: Extends vertically, completing the three dimensions.

Antonio wants to reduce his block of ice from an unspecified volume to exactly 24 cubic feet. To solve this problem, the initial volume of the ice block should be known, or at least approximated. If he reduces the block by shaving off a uniform measurement \( x \) from each dimension, our new calculation describes a volume decreased to \( (l-x) \times (w-x) \times (h-x) = 24 \).

Cubic Equations

Cubic equations appear when dealing with three-dimensional volume problems. They involve terms raised to the third power and can be represented generally as \( ax^3 + bx^2 + cx + d = 0 \).
Understanding cubic equations becomes crucial for tasks like determining how much to reduce each dimension to achieve a desired volume. When reductions affect each dimension equally, the equation \( (a-x)^3 = 24 \) describes the problem when starting with a cube whose sides are initially \( a \). Solving involves finding the value of \( x \) that satisfies the equation.
To tackle cubic equations:

• Use trial and error for simple equations, especially if estimating potential solutions.
• Apply algebraic techniques or computational methods for more precise calculations.

For example, one common approach is guessing close values, checking by substitution, until you find \( x \). With practice, solving cubic equations becomes more intuitive.

Dimensional Reduction

Dimensional reduction refers to reducing one or more dimensions of an object, affecting its volume.
Antonio's goal is a practical application of dimensional reduction. He's tasked with achieving exact changes to his ice block's dimensions to reach a targeted volume. This can often lead to versatile thinking around altering dimensions for specific outcomes.
Key ideas include:

• Understand the original dimensions and calculate any reductions needed.
• Retain uniformity while reducing the size to maintain shape proportion.
• Calculate accurately using algebraic methods to precisely achieve targeted changes.

The essence is in strategically selecting the amount to `shave' off from each dimension, thereby reducing overall dimensions but keeping the object's integrity intact. Balancing these changes while retaining the desired volume in a consistent shape is central to these calculations.