Question

The length \(\ell\) of a box is 3 inches less than the height \(h .\) The width \(w\) is 9 inches less than the height. The box has a volume of 324 cubic inches. Write a model that you can solve to find the length, height, and width of the box.

Short answer

The steps involve first expressing the length and width in terms of the height; then, setting up the equation for the volume in terms of height only; next, solving for the height using this equation, and finally, determining the values for the length and width using the initially defined relationships. The specific numeric values depend on the solution to the cubic equation in step 3.

Step-by-step solution

Express the length and width in terms of the height

From the problem, we know that the length (l) is 3 inches less than the height: \(l = h - 3\). The width (w) is 9 inches less than the height: \(w = h - 9\). Therefore, everything is now expressed in terms of a single variable, h.

Set up the volume equation

The volume \(V\) of a box is given by the product of its length, width, and height: \(V = lwh\). Substituting the expressions for the length and width from step 1 into the volume formula, we get: \(V = (h - 3)(h - 9)h\)

Solve for the height

Substitute the given volume of 324 cubic inches into the volume equation from step 2 to solve for the height: \(324 = (h - 3)(h - 9)h\). This is a cubic equation in h and can be solved by standard algebraic methods, such as factoring, completing the square, or using the cubic formula.

Determine the length and width

Once the height is determined from step 3, use the original relationships to calculate the values for the length and the width. Since length equals height minus 3 and the width equals height minus 9, simply substitute the height value into these two equations to find the other dimensions.

Key concepts

Understanding Algebraic Expressions

When tackling mathematical problems, algebraic expressions are crucial as they represent real-world situations in the language of mathematics. In the context of finding the volume of a box, understanding how to write and manipulate these expressions is foundational.

An algebraic expression is a combination of numbers, variables (symbols representing unknown values), and arithmetic operations like addition and subtraction. In our exercise, we have the algebraic expressions for the length \( \ell = h - 3 \) and the width \( w = h - 9 \), where \( h \) stands for the height of the box, and is the variable we are ultimately solving for. These expressions allow us to describe the dimensions of the box in relation to its height, establishing a relationship between them that is essential for solving the problem at hand.

To improve understanding, always remember to define each variable and explain their roles. The given exercise effectively models real-life scenarios by using algebraic expressions to define the dimensions of an object based on a single variable -- a technique commonly used in algebra to simplify complex problems into more manageable equations.

Demystifying Cubic Equations

In our exercise, once we establish the algebraic expressions for the dimensions of the box, we arrive at a cubic equation. A cubic equation is a type of polynomial equation that involves a variable raised to the third power (cubed). In the form \( ax^3 + bx^2 + cx + d = 0 \), it represents the unknowns within a problem that when solved, reveal precise quantities or measurements.

Our specific equation, \( 324 = (h - 3)(h - 9)h \), is cubic because the variable \( h \) is raised to the third power throughout the product. Solving it requires algebraic methods that may include factoring, graphing, or applying the cubic formula. The goal is to find the value(s) of \( h \) that satisfy the equation, thus providing us with the height of the box. A good tip is to look for patterns or potential factors to simplify the solving process. The cubic nature of the equation implies that there might be one real solution and two complex solutions, or three real solutions, depending on the particular equation parameters. In a practical sense, when dealing with physical objects like a box, we focus on the real, positive solution that makes sense in the given context.

Solving for Variables

Solving for variables is the heart of algebra. It involves finding the value of unknown quantities represented by symbols, typically by isolating the variable on one side of an equation. In the provided exercise, we apply this process to discover the box's dimensions.

After setting up the cubic equation with the box's volume, we use algebraic techniques to solve for \( h \) - which is our variable in this case. Remember, the solution process might include a variety of methods such as simplifying the equation, factoring, or even employing the quadratic formula if the situation is reduced to a quadratic equation after certain manipulations.

Once \( h \) has been found, it's used to determine the length and width using the relationships given in the problem, \( \ell = h - 3 \) and \( w = h - 9 \). As you progress through these steps, ensure each action taken is clear - this not only helps in keeping track of the solving process but also increases comprehension. Particularly for learners, being able to translate a word problem into an algebraic equation and solve for the variables is an essential skill, fostering a deeper understanding of how mathematical concepts apply in various contexts.