What is the volume of a cube with side length [tex]$3 t^8$[/tex]?
Answer (1)
Substitute the side length s = 3 t 5 into the volume formula V = s 3 .
Calculate V = ( 3 t 5 ) 3 .
Simplify the expression to find the volume V = 27 t 15 .
The volume of the cube is 27 t 15 units 3 .
Explanation
Problem Analysis and Given Data We are given a cube with side length s = 3 t 5 units. Our goal is to find the volume of this cube. The formula for the volume of a cube is given by V = s 3 , where s is the side length.
Substitute Side Length into Volume Formula Now, we substitute the given side length s = 3 t 5 into the volume formula:
V = ( 3 t 5 ) 3
To simplify this expression, we need to raise both the constant and the variable term to the power of 3:
V = 3 3 × ( t 5 ) 3
Simplify the Expression We know that 3 3 = 3 × 3 × 3 = 27 . Also, when raising a power to a power, we multiply the exponents: ( t 5 ) 3 = t 5 × 3 = t 15 . Therefore,
V = 27 t 15
Final Answer The volume of the cube is 27 t 15 cubic units. Therefore, the correct answer is 27 t 15 units 3 .
Examples
Understanding the volume of a cube is useful in many real-world scenarios. For example, if you're packing items into a cubic box, knowing the volume helps you determine how much space you have. Similarly, in construction, calculating the volume of concrete needed to fill a cubic mold is essential for accurate material estimation. This concept is also fundamental in fields like architecture and engineering, where spatial calculations are crucial for design and planning.
