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.



The dimensions of the base of Box 2 are $x$ by $4x-1$.



The base area of Box 2 is:

A$4x^2-x$
B$4x-4$
C$5x-1$
D$4x^2-1$

Verified Answer :

  • The dimensions of the base of Box 2 are given as \(x\) and \(4x - 1\).
  • The area of the base of a rectangular box is the product of its length and width: \(\text{Area} = x(4x - 1)\).
  • Expanding the expression, we get \(4x^2 - x\).
  • The correct option is \(\boxed{a}\).

Explanation

  1. Problem Analysis

We are given the dimensions of the base of Box 2 as \(x\) and \(4x - 1\). The problem asks for the base area of Box 2. The area of a rectangle is calculated by multiplying its length by its width. Therefore, we need to multiply the two given dimensions together.

  1. Calculation and Conclusion

To find the base area of Box 2, we multiply its dimensions:
\[ \text{Area} = \text{length} \times \text{width} = x \times (4x - 1) \]
Now, we distribute the \(x\) into the parentheses:
\[ x \times 4x - x \times 1 = 4x^2 - x \]
Comparing this result with the provided options:

A\(4x^2 - x\)
B\(4x - 4\)
C\(5x - 1\)
D\(4x^2 - 1\)

The calculated area matches option (a).

Examples

In manufacturing and design, calculating the area of a base is a fundamental step in determining how much packaging material is needed for a product. For instance, if you are designing a box for cereal, knowing the formula for the base area allows you to calculate the total cardboard surface area required. This basic algebraic expression helps manufacturers optimize material usage, reduce waste, and manage production costs efficiently, which is essential for any business planning to launch a new product.

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