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Chapter 1: Problem 57

Find a formula for the described function and state its domain. An open rectangular box with volume 2 m3 has a square base. Express the surface area of the box as a function of the length of a side of the base.

Short Answer

Expert verified
\( A(x) = x^2 + \frac{8}{x} \) with domain \( x > 0 \).

Step by step solution

01

Understand the Problem

We need to express the surface area of an open rectangular box as a function of the side length of its square base, given that the box has a volume of 2 m鲁.
02

Define Variables

Let the side length of the square base be denoted by \( x \) meters. Since the base is square, both sides are \( x \). Let the height of the box be represented by \( h \) meters.
03

Use Volume Formula

The volume \( V \) of the box is given by the formula \( V = x^2h \). Since the volume is 2 m鲁, we have the equation \( x^2h = 2 \).
04

Express Height (h) in Terms of x

Solve the equation \( x^2h = 2 \) for \( h \). This gives \( h = \frac{2}{x^2} \).
05

Write Surface Area Formula

The surface area \( A \) of the open box consists of the base and the four sides. Since the box has a square base side \( x \) and height \( h \), the surface area is \( A = x^2 + 4xh \).
06

Substitute Height in Surface Area Formula

Substitute \( h = \frac{2}{x^2} \) into the surface area formula \( A = x^2 + 4xh \). This results in \( A = x^2 + 4x\left(\frac{2}{x^2}\right) \). Simplifying gives \( A = x^2 + \frac{8}{x} \).
07

State the Function and Domain

The function for surface area as a function of base side length \( x \) is \( A(x) = x^2 + \frac{8}{x} \). The domain of the function is \( x > 0 \) because the side length of the box must be positive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume of a Box
The volume of a box measures how much space the box encloses or how much it can hold. For a rectangular box, volume is found by multiplying the area of the base by the height of the box. If the base is a square with side length \( x \), then the area of the base is \( x^2 \). Thus, the volume \( V \) is given by the formula:
  • \( V = x^2h \)
where \( h \) is the height of the box.
Since we know the volume of the box is 2 m鲁, we use this formula to find the relationship between \( x \) and \( h \). Solving for the height gives us \( h = \frac{2}{x^2} \), showing that as the side of the base changes, so does the height, maintaining constant volume.
Function of One Variable
A function of one variable is a mathematical expression where the value of one variable depends on the value of another. In this case, we want to express the surface area \( A \) of a box as a single function that depends on the base side length \( x \).To achieve this:
  • We first define the relationship between the side of the base and the rest of the box's dimensions.
  • Next, we integrate these relationships into the formula for surface area \( A \).
  • We substitute any variable expressions to reflect only \( x \).
This results in our surface area function: \( A(x) = x^2 + \frac{8}{x} \). This expression allows us to quickly calculate the surface area for any given side length \( x \).
Domain of a Function
The domain of a function refers to all possible input values that let the function produce real and meaningful output values. For our surface area function \( A(x) = x^2 + \frac{8}{x} \), the input value is the side length \( x \) of the box's base. To determine the domain:
  • We consider the nature of \( x \) as a real-world measurement. It represents a length and must be positive, ensuring that the box dimensions are physically valid.
  • We note that \( x \) cannot be zero because this would result in undefined mathematical operations, such as division by zero in the term \( \frac{8}{x} \).
Thus, the domain is expressed as all positive \( x \), or \( x > 0 \).
Square Base
The notion of a square base plays a crucial role in the geometry of a box, especially in calculations involving surface area and volume. A square base means both dimensions of the box's base are equal, simplifying calculations.
  • Each side of the square base is represented by \( x \).
  • The area of the base is \( x^2 \), a straightforward calculation due to the equal sides.
  • This uniformity influences the formulas we derive, such as the expression for volume \( V = x^2h \) and surface area \( A = x^2 + 4xh \).
Understanding these properties of a square base allows us to easily manipulate formulas and better visualize the physical box in three dimensions.

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Most popular questions from this chapter

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