Problem 4 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The volume is 336 cubic units.

Step 1: 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.

Step 2: 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 \( V = 14 \times 8 \times 3 \).

Step 3: 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 \( V = 336 \).

Step 4: Conclude the Calculation

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

Key Concepts

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.