Problem 34 - Contemporary Precalculus

Question

A box with a square base of side \(x\) is four times higher than it is wide. Express the volume \(V\) of the box as a function of \(x\)

Step-by-Step Solution

Answer

Question: Express the volume of a box with a square base of side length x and height four times the width as a function of x. Answer: V(x) = 4x^3

Step 1: Determine the height of the box

Since the height of the box is four times higher than its width, and the width is given as \(x\), we can express the height as \(4x\).

Step 2: Use the volume formula for a rectangular box

The formula for the volume of a rectangular box is \(V = L \times W \times H\) where V is the volume, L is the length, W is the width, and H is the height.

Step 3: Substitute the given values in the volume formula

We know that the base length and width are equal, so the length is also \(x\). And from step 1, the height is \(4x\). Applying these values to the volume formula, we get: \(V = x \times x \times 4x\)

Step 4: Simplify the expression

We can simplify the above expression by multiplying them together: \(V = 4x^3\) So, the volume of the box as a function of \(x\) is \(V(x) = 4x^3\).

Key Concepts

Volume Formula

The volume formula is essential when solving problems related to the size of geometric shapes. For a rectangular box, the volume is calculated using the formula:

• \[ V = L \times W \times H \]

Here, \(L\) represents the length, \(W\) is the width, and \(H\) is the height of the box. Volume, denoted as \(V\), measures the three-dimensional space occupied by the box. All these dimensions must be in the same units to obtain an accurate volume measurement. Understanding this formula is vital for applying it to various geometric problems. This particular formula simplifies to many specific cases depending on the values of length, width, and height.

Function of a Variable

A function of a variable describes how various quantities depend on a changing input. In mathematics, functions provide us a way of predicting one quantity based on others. This often involves expressing relationships through equations. Here, our function of interest is \(V(x) = 4x^3\), where \(V\) is dependent on the variable \(x\), the side length of the square base.
This function allows you to compute the volume for any given side length of the base, \(x\). Functions make it possible to understand dynamic changes in shapes when one dimension varies, like how volume changes as the side length of a rectangular box changes.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents. These types of functions often describe a wide variety of real-world situations, including in geometry. The function \(V(x) = 4x^3\) is a polynomial of degree three, indicating a cubic relationship between the box's base length \(x\) and its volume.

• In this polynomial:
  • \(x^3\) shows the cubic relationship dictated by multiplying the dimensions.
  • The coefficient \(4\) reflects how the height affects the overall volume.
• Polynomials like this can represent surface areas, volumes, trajectories, and other measurable attributes in mathematics.

Understanding polynomial behavior helps predict outcomes in geometric applications and solve complex problems by breaking them down into these mathematical expressions.

Geometric Applications

Geometric applications involve using mathematical principles to solve real-world problems involving shapes and forms. In our problem, the application centers around calculating the volume of a rectangular box with constraints about its dimensions. Here’s how you can visualize and apply the concepts:

• A box with variable base sides where one dimension grows proportionally taller creates an opportunity to understand spatial scaling.
• Application of geometry involves setting equations based on given conditions, like how the height is four times the width.
• Mathematics provides tools to model such problems, simplify them, and extract meaningful insights, such as calculating resources needed for production based on volume.

These applications span a variety of fields from architecture to engineering, where space must be precisely defined and utilized.