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Chapter 4: Problem 84

If the value of \(x\) is \(6,\) what is the volume of the cube (in cubic units)?

Short Answer

Expert verified
The volume of the cube is 216 cubic units.

Step by step solution

01

Understand the problem

A cube has all sides of equal length. To find the volume of the cube, use the formula for the volume of a cube.
02

Recall the volume formula

The volume of a cube is given by the formula: \[ V = s^3 \] where \( s \) is the length of one side of the cube.
03

Substitute the value of \( s \)

Given that \( x = 6 \), and since the side length \( s \) is equal to \( x \), substitute \( s = 6 \) into the volume formula: \[ V = 6^3 \]
04

Calculate the volume

Calculate \( 6^3 \) to find the volume: \[ 6^3 = 6 \times 6 \times 6 = 216 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube Volume Formula
To understand how to find the volume of a cube, it's essential to use the correct formula. The volume of a cube is calculated by raising the length of one of its sides (denoted as 's') to the power of three. The formula is straightforward: \[ V = s^3 \]Here, 'V' represents the volume of the cube, and 's' is the length of one side. This formula works because a cube's volume is simply the space it occupies, which is found by multiplying its length, width, and height. Since all sides of a cube are equal, you multiply the side length three times.
Exponents
Exponents are a shorthand way of showing repeated multiplication. When a number is raised to an exponent, it means you multiply that number by itself a certain number of times. For instance, in the cube volume formula \( s^3 \), the exponent 3 implies we multiply the side length 's' by itself three times. In the example given in the exercise, where the side length \( s = 6 \), the calculation would be \[ 6^3 = 6 \times 6 \times 6 \] which equals 216. Exponents are a powerful mathematical tool that simplifies what could otherwise be lengthy multiplication.
Substitution of Values
Substitution is a crucial concept in mathematics where you replace a variable with its actual value. In this exercise, we're given that \( x = 6 \) and need to find the volume of the cube. Knowing the formula \[ V = s^3 \] and that the side length \( s \) is equal to \( x \), we substitute \( s = 6 \) into the formula. This substitution simplifies the problem to \[ V = 6^3 \] By performing this substitution, we convert a general formula into a specific calculation, making it possible to compute the cube's volume. Finally, calculating \( 6^3 = 216 \), we find that the volume of the cube is 216 cubic units.

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Most popular questions from this chapter

Add or subtract as indicated. \((-5 t+13 s)+(8 t-3 s)\)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ 5 k^{2}\left(k^{3}-3\right)\left(k^{2}-k+4\right) $$

Find each product. $$ -5 r(r+1)^{3} $$

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 30 \frac{1}{3} \times 29 \frac{2}{3} $$

Perform each indicated operation. \(\left[\left(8 m^{2}+4 m-7\right)-\left(2 m^{2}-5 m+2\right)\right]-\left(m^{2}+m+1\right)\)
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