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.

Dimensions: $x$ by $3 x$ by $x^3$

Area of base $= x(3 x) = 3 x^2$



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

Dimensions: $x$ by $4 x - 1$ by $x^3$

Area of base $= x(4 x - 1) = 4 x^2 - x$



Complete the statements about the number of terms in the polynomial representing the volume of each box.



Box 1's volume is modeled by a □, so it will be a □.

Box 2's volume is modeled by a □, so it will be a □.

Verified Answer :

  • The volume of a rectangular box is the product of its length, width, and height.
  • For Box 1, the volume is \(V_1 = x \cdot 3x \cdot x^3 = 3x^5\), which is a monomial because it has one term.
  • For Box 2, the volume is \(V_2 = x \cdot (4x - 1) \cdot x^3 = 4x^5 - x^4\), which is a binomial because it has two terms.
  • Final answer: Box 1's volume is modeled by \(\boxed{3x^5}\) (monomial), and Box 2's volume is modeled by \(\boxed{4x^5 - x^4}\) (binomial).

Explanation

  1. Analysis of Box Dimensions and Volume Formula

The volume \(V\) of a rectangular box is calculated using the formula \(V = \text{length} \times \text{width} \times \text{height}\). We are given the dimensions for two different box designs in terms of the variable \(x\).
Box 1 has dimensions \(x\), \(3x\), and \(x^3\).
Box 2 has dimensions \(x\), \(4x - 1\), and \(x^3\).

  1. Volume Calculation and Classification for Box 1

To find the volume of Box 1, we multiply its dimensions:
\[V_1 = x \times (3x) \times (x^3)\]
Simplifying this expression:
\[V_1 = 3 \times x \times x \times x^3 = 3x^5\]
The volume \(3x^5\) consists of a single term. A polynomial with one term is called a monomial.

  1. Volume Calculation and Classification for Box 2

To find the volume of Box 2, we multiply its dimensions:
\[V_2 = x \times (4x - 1) \times (x^3)\]
Simplifying this expression by distributing \(x^4\) through the parentheses:
\[V_2 = x^4(4x - 1) = 4x^5 - x^4\]
The volume \(4x^5 - x^4\) consists of two terms. A polynomial with two terms is called a binomial.

  1. Conclusion and Final Answer

Based on the calculations above:

  • Box 1's volume is modeled by a \(3x^5\), so it will be a monomial.
  • Box 2's volume is modeled by a \(4x^5 - x^4\), so it will be a binomial.

Thus, the completed statements are:
Box 1's volume is modeled by a \(\boxed{3x^5}\), so it will be a \(\boxed{\text{monomial}}\).
Box 2's volume is modeled by a \(\boxed{4x^5 - x^4}\), so it will be a \(\boxed{\text{binomial}}\).

Examples

In manufacturing and packaging design, understanding the volume of a container is crucial. For instance, if a company wants to design a cereal box, the volume tells them how much product will fit inside. By expressing this volume as a polynomial, engineers can easily calculate how changing dimensions (like increasing the width \(x\)) will affect the total volume, helping them optimize space and material usage efficiently.

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