Recall the formula for the volume of a cube: [tex]V=s^3[/tex].
The side length, s, can be found using the equation [tex]3 s=729[/tex].
Which of the following statements are true?
A. This is a perfect cube.
B. The side length is 243 centimeters.
C. The side length is 9 centimeters.
D. Taking the cube root of the volume will determine the side length.
E. If you multiply the volume by three, you can determine the side length.
Answer (2)
The side length of the cube is 243 centimeters, and the volume is 14,348,907 cubic centimeters. True statements include that this is a perfect cube, the side length is 243 centimeters, and taking the cube root of the volume gives the side length. The statements about the side length being 9 centimeters and multiplying the volume by three to determine the side length are incorrect.
;
Solve the equation 3 s = 729 to find the side length s = 243 .
Calculate the volume using V = s 3 , which gives V = 24 3 3 = 14348907 .
Verify that taking the cube root of the volume gives the side length: 3 14348907 = 243 .
Conclude that the side length is 243 cm and the volume is 14348907 cubic centimeters. 243
Explanation
Understanding the Problem We are given the formula for the volume of a cube, V = s 3 , where s is the side length. We are also given the equation 3 s = 729 . We need to find the side length s and analyze the given statements.
Solving for Side Length First, let's solve the equation 3 s = 729 for s . To do this, we divide both sides of the equation by 3: s = 3 729 = 243
So, the side length s is 243 centimeters.
Calculating the Volume Now, let's calculate the volume V of the cube using the formula V = s 3 . Since s = 243 , we have: V = ( 243 ) 3 = 243 × 243 × 243 = 14348907
So, the volume of the cube is 14,348,907 cubic centimeters.
Analyzing the Statements Now, let's analyze the given statements:
"The side length is 243 centimeters." This statement is correct based on our calculation.
"The side length is 9 centimeters." This statement is incorrect.
"Taking the cube root of the volume will determine the side length." Let's check this: 3 V = 3 14348907 = 243 . This statement is correct.
"If you multiply the volume by three, you can determine the side length." Let's check this: 3 V = 3 × 14348907 = 43046721 . This is not equal to the side length 243. So, this statement is incorrect.
Final Answer Therefore, the side length of the cube is 243 centimeters.
Examples
Cubes are fundamental shapes in geometry and have many real-world applications. For example, many boxes are made in the shape of cubes or rectangular prisms, and calculating their volume is essential for determining how much they can hold. In architecture, understanding the volume and surface area of cubes is important for designing buildings and structures. In manufacturing, many products are packaged in cube-shaped containers for efficient storage and transportation. Understanding the relationship between a cube's side length and its volume is a basic concept with wide-ranging practical applications.
