Problem 47 - Precalculus : Building Concepts and Connections

Question

A rectangular solid has a square base and a height that is 2 inches less than the length of one side of the base. What lengths of the base will produce a volume greater than or equal to 32 inches?

Step-by-Step Solution

Answer

The lengths of the base that will produce a volume greater than or equal to 32 inches are all lengths greater than or equal to 4 inches.

Step 1: Set up the equation in terms of length

The volume of the solid is given by the formula volume = length * width * height. The width is equal to the length because the base is square, and the height is two inches less than the length, so volume = length * length * (length - 2). After simplification of the equation, we will get: volume = \(length^3 - 2length^2\).

Step 2: Equation Evaluation

The volume needs to be equal to or greater than 32 inches. Substituting into the equation, we will have: \(32 \leq length^3 - 2length^2\). We can rewrite it as : \(length^3 - 2length^2 - 32 \geq 0 \).

Step 3: Solve the equation for length

This equation can be solved by using a factoring method or quadratic formula. However, this is a cubic equation, it can also be solved by trying up some values. On trying some values for length, we find that length = 4 makes the equation equal to zero. This means that any length greater than 4 will make the cubic equation positive (since the leading coefficient for \(length^3\) is positive), hence the volume greater than 32.

Key Concepts

Cubic Equations

In mathematics, cubic equations are polynomials of degree three. They often take the form: \( ax^3 + bx^2 + cx + d = 0 \), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\). These equations involve solving for \(x\), the variable, which can have up to three real roots. Cubic equations are interesting because they can represent a variety of problems in real life, such as calculating volumes. For instance, the problem of finding the volume of a rectangular solid with sides defined by algebraic expressions can be solved using cubic equations because they naturally include terms with \(x^3\).
Here, the expression for the volume of the rectangular solid is \( length^3 - 2length^2 \), which represents a classic cubic equation once set equal to the constraints or conditions given in a problem, like having a volume greater than or equal to 32. Understanding cubic equations allows one to manipulate them using algebraic techniques and evaluate potential solutions that satisfy these inequalities. Knowing how to solve them showcases a fundamental algebraic skill necessary for more complex mathematics.

Factorization

Factorization involves breaking down expressions into products of simpler factors. This is an essential algebraic process, especially when dealing with polynomials, as it helps to simplify and solve equations. To factorize a cubic equation, one might first look for common factors or employ techniques like trial and error to find rational roots.
In the given problem, the equation \( length^3 - 2length^2 - 32 = 0 \) can potentially be solved by factorization. By trying different values for the variable 'length,' we can determine the suitable factor, i.e., \( length = 4 \). After establishing a root, such as \(x = 4 \), the polynomial can be divided by \( (x - 4) \), simplifying it to decipher other factors, if any.

• This technique helps discover integer solutions efficiently.
• It reduces complexity by breaking down higher-degree polynomials into manageable quadratic or linear factors.

Being able to factorize is crucial in problem-solving, allowing you to identify and work with viable roots and simplify expressions, which is especially helpful in solving problems that involve cubic equations.

Inequalities

Inequalities involve statements that indicate one quantity is larger or smaller than another. Solving inequalities requires understanding of how to manipulate algebraic equations while maintaining the inequality's direction. They are ubiquitous in mathematics whenever a range of solutions rather than a singular answer is possible.
In this context, the exercise asks for values of 'length' such that the volume of the solid is at least 32 cubic inches, corresponding to the inequality \( length^3 - 2length^2 - 32 \geq 0 \). This inequality suggests that we need to find values of 'length' where this expression is zero or positive. Using solutions from factorization or other methods, we can identify critical points where inequality holds true or changes sign.

• Understand critical points and checking intervals between them.
• Graphing the expression, if possible, provides a clear visual of where the inequality holds.

Solving inequalities involves comprehensive steps but is highly beneficial for a fuller understanding of the solution space of a given problem, especially in terms of defining ranges for possible solutions rather than singular outcomes.