Problem 43 - Algebra 1

Question

Write an equation or an inequality to model the real-life situation. The volume \(V\) of a cube with a side length \(s\) is less than or equal to thirty minus three.

Step-by-Step Solution

Answer

The inequality that models the given real-life situation is \(s^3 \leq 27\).

Step 1 Recall The Formula for Cube Volume

Remember that the formula for the volume of a cube is \(V = s^3\). Here \(V\) is the volume and \(s\) the side length of the cube.

Step 2 Understand the Given Conditions

The condition mentioned is that the volume of the cube is less than or equal to thirty minus three, which we can write as \(30 - 3\) or 27.

Step 3 Write the Inequality

Given the formula for volume and the condition, we can then write the inequality: \(s^3 \leq 27\).

Key Concepts

Volume of a Cube

When it comes to understanding the concept of volume for three-dimensional shapes, the cube is one of the simplest forms to start with. The volume of a cube is found by multiplying the length of one of its edges, known as the side length, by itself three times. This is captured by the algebraic expression \( V = s^3 \).

In a real-life context, knowing the volume of a cube can help you determine how much space it occupies, which is useful for storage, shipping, and many other practical situations. The key here is that all sides are equal in length, and this uniformity leads to the simple calculation. To really get to grips with this concept, visualize filling a cube with small unit cubes, say 1 cubic centimeter each, and counting the total to find the volume.

Algebraic Inequality

Now, let's delve deeper into the concept of an algebraic inequality. Unlike equations, inequalities don't suggest that two expressions are equal; instead, they show a relationship where one is greater than or less than the other.

An algebraic inequality, like \( s^3 \leq 27 \), tells us that the cube of the side length \( s \) is no greater than 27. It represents a range of values, rather than a single number. This concept extends to scenarios where we have limitations or restrictions in real-life situations. For example, suppose you have a container that can hold no more than 27 cubic units; an inequality would help you understand the maximum size of an item that could fit.

Solving Inequalities

When faced with an inequality, like \( s^3 \leq 27 \), solving it means finding all possible values of the variable that make the inequality true. Here are some basic steps to keep in mind:

• Perform the same operations on both sides of the inequality.
• Be mindful of the direction of the inequality sign.
• Remember that multiplying or dividing by a negative number reverses the inequality sign.

To solve \( s^3 \leq 27 \) specifically, we would determine the cube root of 27 to find the maximum value that \( s \) could be. Understanding how to resolve such inequalities is crucial for tasks ranging from optimizing areas in design to setting limits in scientific experiments.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They are fundamental in conveying mathematical concepts in a concise manner.

Consider our cube volume formula \( V = s^3 \). It compactly expresses how the volume \( V \) varies with changes in the side length \( s \). The expression does not have an equality sign unless it is part of a specific equation or inequality being solved. Mastery of algebraic expressions is vital for students as it forms the foundation for understanding more complex mathematical ideas and for describing relationships in fields like physics, economics, and beyond.