Problem 63 - Calculus with Concepts in Calculus

Question

Express the surface area \(S\) of a rectangular box with no top as a function of the dimensions \(x, y\), and \(z\).

Step-by-Step Solution

Answer

S(x, y, z) = xy + 2xz + 2yz.

Step 1: Understanding the Problem

We need to find an expression for the surface area of an open-top rectangular box. The box has three dimensions: length \(x\), width \(y\), and height \(z\). Since the box has no top, we only consider the bottom and the four sides.

Step 2: Identifying the Surfaces

The box consists of five surfaces: one bottom and four sides. The bottom is a rectangle with area \(xy\). There are two side surfaces with dimensions \(x\) and \(z\), and two other side surfaces with dimensions \(y\) and \(z\).

Step 3: Calculating the Surface Area of Each Part

Calculate the area of each piece:- The bottom has area \(xy\).- Each of the two larger side rectangles (with dimensions \(x\) and \(z\)) has area \(xz\).- Each of the two smaller side rectangles (with dimensions \(y\) and \(z\)) has area \(yz\).

Step 4: Combining All Surface Areas

The total surface area \(S\) is the sum of all these areas:\[ S = xy + 2xz + 2yz \]

Step 5: Final Function Expression

The surface area \(S\) as a function of the dimensions \(x\), \(y\), and \(z\) is given by:\[ S(x, y, z) = xy + 2xz + 2yz \]

Key Concepts

surface area

Surface area refers to the total area that the surface of a three-dimensional object occupies. It's like trying to calculate how much wrapping paper you would need to cover the entire object completely. For the surface area of a shape, we add up the areas of all its faces or surfaces.
For our open-top rectangular box, we identify five surfaces that contribute to the surface area:

• The base (or bottom) of the box with an area calculated by multiplying its length and width, which is \(xy\).
• The two larger side faces have dimensions of \(x\) by \(z\), and so each has an area of \(xz\).
• Similarly, the two smaller side faces have dimensions of \(y\) by \(z\), with each having an area of \(yz\).

We aim to obtain the total surface area by summing these individual areas. This allows us to write the expression:\[S = xy + 2xz + 2yz\]
Understanding this expression is crucial because it shows how each dimension plays a role in the overall surface area.

rectangular box

A rectangular box, also often referred to as a rectangular prism, is a 3D shape that features six rectangle-shaped faces. These could be of equal or different lengths, resulting in a box that is not always cubical.
For our specific problem, we have a rectangular box where the dimensions are given by the length \(x\), width \(y\), and height \(z\). Importantly, our box is open at the top, meaning it only has five surfaces or faces to consider when calculating the surface area.
This concept is foundational as it illustrates how the different elements of the box contribute to the overall geometry and physical properties such as surface area. Understanding what a rectangular box is helps break down complex problems into understandable components, reflecting real-world objects like cartons and storage boxes.

dimensions

Dimensions are the measurements that define the size of an object in space. For any rectangular shape, these typically refer to length, width, and height.
In this exercise, dimensions are denoted by \(x\), \(y\), and \(z\). The dimensions dictate not only the shape of the box but fundamentally influence calculations such as volume and surface area.
For an open-top box, knowing its dimensions allows us to compute the areas of the bottom and the side surfaces. Each dimension is uniquely paired with another to form a specific face's area:

• Length \(x\) combined with width \(y\) determines the area of the bottom \(xy\).
• A length \(x\) and height \(z\) provide the areas of two opposite side faces, each \(xz\).
• A width \(y\) and a height \(z\) give the areas of the other two opposite side faces, each \(yz\).

This highlights that dimensions are more than just numbers; they are core to the geometric and functional characteristics of the box.