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Chapter 2: Problem 4

Substitute the given values into each given formula and solve for the unknown variable. $$ V=l w h ; \quad l=14, w=8, h=3 $$

Short Answer

Expert verified
The volume is 336 cubic units.

Step by step solution

01

Identify the Formula

Recognize that the formula provided is the volume of a rectangular prism, which is given by \( V = lwh \). Here, \( V \) is the volume, \( l \) is the length, \( w \) is the width, and \( h \) is the height.
02

Substitute Given Values into the Formula

Replace the variables \( l \), \( w \), and \( h \) in the formula with their given values. Thus, substitute \( l = 14 \), \( w = 8 \), and \( h = 3 \) into the equation, resulting in \( V = 14 \times 8 \times 3 \).
03

Perform the Multiplication

Calculate the volume by multiplying the values together: \( 14 \times 8 = 112 \), then multiply the result by the height: \( 112 \times 3 = 336 \). Thus, the volume \( V = 336 \).
04

Conclude the Calculation

The calculated value represents the volume of the rectangular prism using the given dimensions. Therefore, the volume is 336 cubic units.

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Key Concepts

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

Substitution in Algebra
Substitution in algebra is a helpful technique where you replace variables in an equation with specific known values. This makes the equation simpler to solve. In our problem, the formula for the volume of a rectangular prism is given: \[ V = lwh \]Here, each letter represents a different dimension of the prism. The task involves substituting the values given for length (\( l = 14 \)), width (\( w = 8 \)), and height (\( h = 3 \)) into this formula.
  • Start by identifying which values you have and what they represent.
  • Plug these values into the equation in the place of their respective variables.
  • This substitution transforms your algebraic equation into a numerical expression: \( V = 14 \times 8 \times 3 \).
This process lays the groundwork for further arithmetic calculations, allowing us to solve for the unknown variable, in this case, the volume \( V \).
Calculation Steps
Once we have substituted the values, the next step is to carefully carry out the calculations. Breaking down these steps ensures that you arrive at the correct solution:
  • First Multiply: Calculate \( 14 \times 8 \). This step involves a straightforward multiplication of two numbers, resulting in \( 112 \).
  • Second Multiply: Now, take that result (\( 112 \)) and multiply it by the height, \( 3 \), which gives \( 112 \times 3 = 336 \).
Each step involves a clear and careful multiplication to maintain accuracy. Often, it's beneficial to pause between each arithmetic step to double-check your math. Breaking down the calculation like this not only helps error-checking but also aids in understanding how the multiplication interacts in the context of the formula.
Multiplication
Multiplication is a fundamental arithmetic operation that combines equal groups. In finding the volume of a rectangular prism, you multiply the three dimensions: length, width, and height. Each value represents a number of unit lengths in the respective dimensions:
  • Length (\( l \)): Represents how far the object extends in one dimension.
  • Width (\( w \)): Provides a measure of the object's breadth, or span, in a perpendicular direction to length.
  • Height (\( h \)): Indicates the extent of the object in a vertical direction, creating a three-dimensional space.
The calculation of \( 14 \times 8 \times 3 \) involves multiplying length by width, then taking that result and multiplying by height. This process combines all three dimensions, giving us a comprehensive measure of volume, which is the amount of space the prism occupies. Understanding multiplication in this context shows it as a tool for three-dimensional measurements.

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