Problem 73 - Algebra A Combined Function

Question

The function \(V(x)=x^{3}\) may be used to find the volume of a cube if we are given the length \(x\) of a side. Use this function to answer Exercises 73 and 74 Find the volume of a cube whose side is 14 inches.

Step-by-Step Solution

Answer

The volume of the cube is 2744 cubic inches.

Step 1: Identify the given length

The problem states that the side of the cube is 14 inches. Therefore, \( x = 14 \).

Step 2: Recall or Write the volume function

The volume of a cube can be found using the function \( V(x) = x^3 \), where \( x \) represents the side length.

Step 3: Substitute the side length into the function

Substitute \( x = 14 \) into the volume function: \( V(14) = 14^3 \).

Step 4: Calculate the volume

Compute the cube of 14: \( 14^3 = 14 \times 14 \times 14 = 2744 \). Thus, the volume of the cube is 2744 cubic inches.

Key Concepts

Volume Formula

When it comes to understanding the concept of volume, particularly for geometric shapes like cubes, one of the most fundamental formulas we use is the volume formula. For a cube, the formula to calculate its volume is given by:\[ V = x^3 \]Where:

• \( V \) represents the volume of the cube
• \( x \) is the length of each side of the cube

This formula is derived from the fact that a cube is a special case of a rectangular prism where all sides have equal length. Calculating the volume means finding out how much space is inside the 3D shape. The volume formula for a cube is particularly straightforward and is a great example of how dimensions relate to volume. This simplicity makes it easy to apply in many practical scenarios.

Cube Dimensions

Understanding cube dimensions is essential for using the volume formula correctly. A cube is a three-dimensional shape with equal length on all sides, meaning all three dimensions—length, width, and height—are the same. In mathematical terms, if each side of the cube is of length \( x \), then:

• Length = \( x \)
• Width = \( x \)
• Height = \( x \)

The fact that these dimensions are identical simplifies many mathematical calculations, including volume. Moreover, the symmetry and uniformity of cube dimensions make understanding its geometry easier, providing a clearer picture when visualizing 3D objects. So, when calculating volume, we don't need separate numbers for each dimension; the same value works for all three.

Exponentiation

Exponentiation is a mathematical operation that lets us express repeated multiplication compactly. In the context of finding the volume of a cube, we often see cube calculations, which involve raising a number to the power of three. This is expressed using the exponent of 3, noted in our formula:\[ x^3 \]Here, exponentiation means multiplying \( x \) by itself twice more. For example, with a side length of 14 inches:\[ 14^3 = 14 \times 14 \times 14 = 2744 \]Exponentiation is crucial in volume calculations as it reflects the three-dimensional aspect of shapes like cubes. Understanding how to perform exponentiation ensures you can calculate large numbers efficiently and grasp how dimensions scale in 3D space. It's a handy tool not only for cube volume but also across various fields requiring mathematical computation.