Answered

Celine's Cereal Company is launching a new brand of cereal and she is considering two different sizes for the base of the boxes.



Box 1: The length is 3 times the width.

Box 2: The length is 1 less than 4 times the width.

Dimensions of Box 2: $x$ by $4 x-1$ by $x^3$



The volume of Box 2 is given by:

A$4 x^6-x^3$
B$4 x^5-x^4$
C$4 x^5+x^4$
D$x^3+5 x-1$

Verified Answer :

  • Identify dimensions: \(x\), \((4x-1)\), and \(x^3\).
  • Set up volume formula: \(V = x(4x-1)x^3\).
  • Multiply dimensions: \(V = (4x^2 - x)x^3 = 4x^5 - x^4\).
  • Match with options: The correct volume is \(4x^5 - x^4\), which is option \(\boxed{b}\).

Explanation

  1. Understanding the Problem and Dimensions

The problem asks us to determine the volume of a rectangular cereal box, identified as 'Box 2'. The dimensions of this box are given as \(x\), \(4x-1\), and \(x^3\). The volume \(V\) of a rectangular box is calculated by multiplying its three dimensions: length, width, and height. In this case, the volume is given by the product:
\[V = x \cdot (4x - 1) \cdot x^3\]

  1. Calculating the Volume of Box 2

To calculate the volume, we first multiply the first two dimensions, \(x\) and \((4x - 1)\):
\[x \cdot (4x - 1) = x \cdot 4x - x \cdot 1 = 4x^2 - x\]
Next, we multiply this intermediate result by the third dimension, \(x^3\):
\[V = (4x^2 - x) \cdot x^3\]
Distributing \(x^3\) into the expression:
\[V = (4x^2 \cdot x^3) - (x \cdot x^3)\]
Using the laws of exponents, where \(x^a \cdot x^b = x^{a+b}\):
\[4x^2 \cdot x^3 = 4x^{2+3} = 4x^5\]
\[x \cdot x^3 = x^1 \cdot x^3 = x^{1+3} = x^4\]
Thus, the volume is:
\[V = 4x^5 - x^4\]

  1. Final Answer Selection

Comparing the calculated volume \(V = 4x^5 - x^4\) with the given options:

A\(4x^6 - x^3\)
B\(4x^5 - x^4\)
C\(4x^5 + x^4\)
D\(x^3 + 5x - 1\)

The calculated expression matches option (b). Therefore, the correct volume is \(4x^5 - x^4\), and the final answer is \(\boxed{b}\).

Examples

In manufacturing and packaging, determining the volume of containers is a fundamental task to ensure products fit properly and to optimize material usage. By expressing dimensions as algebraic variables, engineers can quickly calculate the volume for different box sizes as the variable \(x\) changes, allowing them to make informed decisions about product design and distribution. This simple polynomial multiplication is a basic but essential skill in logistics, supply chain management, and industrial engineering.

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