Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Volume of a Box An open-top box is made from a sheet of metal by cutting squares from each corner and folding up the sides. The sheet has a length of 19 inches and a width of 13 inches. If \(x\) is the length of one side of each square to be cut out, write a function, \(V(x),\) for the volume of the box in terms of \(x\).

Short answer

\[ V(x) = 4x^3 - 64x^2 + 247x \]

Step-by-step solution

- Define Variables

Let’s define the variable: let the length of the square cut from each corner be denoted as x. The original sheet is 19 inches in length and 13 inches in width.

- Determine New Dimensions

After cutting out squares from each corner of length x, the new dimensions of the sheet when folded up to form the box will be as follows:New length: 19 - 2x New width: 13 - 2xHeight of the box: x

- Calculate the Volume

The volume of a box is given by the product of its length, width, and height. Therefore, the volume V(x) of the box can be written as:\[ V(x) = (19 - 2x)(13 - 2x)x \]

- Simplify the Equation

Expand and simplify the equation to obtain the volume function in terms of x:\[ V(x) = x(19 - 2x)(13 - 2x) \]First, expand the terms inside the parentheses:\[ = x(247 - 26x - 38x + 4x^2) \]Combine like terms:\[ = x(247 - 64x + 4x^2) \]Finally, distribute x:\[ V(x) = 4x^3 - 64x^2 + 247x \]

Key concepts

volume calculation

In mathematics, the volume of a shape is the amount of space it occupies in three dimensions. For a box, calculating the volume means determining how much space the box can hold inside it. This is typically measured in cubic units. For our problem, we have a box constructed by folding a metal sheet, so we need to derive the volume from its adjusted dimensions.
Volumes of rectangular objects can be easily found using the formula:
Volume, V = length × width × height
For this box, we will consider its new dimensions after cutting out squares and folding up the sides. The critical part is to express the volume as a function of the cutout length, forming an algebraic representation for deeper analysis.

polynomial functions

Polynomial functions are algebraic expressions that involve sums of multiple terms consisting of variables raised to different powers and their coefficients. A key trait of these functions is their versatility in graphing and modeling real-world scenarios, like the volume of our box.
In our problem, we found an expression for the volume in terms of the side length of the square cut-out, denoted by x, resulting in the polynomial function:
\[ V(x) = 4x^3 - 64x^2 + 247x \]
This is a cubic polynomial function because the highest power of x is 3. Understanding polynomial functions helps in analyzing properties such as finding critical points, which are useful for maximizing or minimizing volume in practical scenarios.

algebraic expressions

An algebraic expression includes numbers, variables, and operations. These expressions can represent various quantities and relationships in mathematics. In our exercise, we used algebraic expressions to define the new dimensions and the volume of the box. Let's break it down:
Initially, the length and width of the metal sheet are given. By cutting out squares of side x from each corner and folding the sides up, the new dimensions become:

• New length: 19 - 2x
• New width: 13 - 2x
• Height: x

Combining these measurements, we formulated the volume expression as:
\[ V(x) = x(19 - 2x)(13 - 2x) \]
Further simplifying achievements algebraically refined our understanding and provided a clear volume polynomial.
This shows the power of algebraic expressions in turning a geometric problem into a solvable equation.

problem-solving steps

Solving mathematical problems involves a sequence of logical steps. Using our volume example, let's outline this process clearly:
Step 1: Define Variables
We start by identifying and defining the key variables involved. Here, we denote x as the length of each side of the cut-out square.
Step 2: Determine New Dimensions
Next, we calculate how the original dimensions of the metal sheet alter after cutting out squares. The new length and width become (19 - 2x) and (13 - 2x) respectively, while the height becomes x.
Step 3: Calculate the Volume
Using the volume formula, we formally express the volume in terms of x:
\[ V(x) = x(19 - 2x)(13 - 2x) \]
Step 4: Simplify the Equation
Finally, we expand and simplify the product to express the volume as a polynomial function:
\[ V(x) = 4x^3 - 64x^2 + 247x \]
Breaking down each problem-solving step systematically ensures clarity and deepens understanding, making even complex problems manageable.