Problem 27 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Express the surface area of a cube as a function of the length \(s\) of one side.

Step-by-Step Solution

Answer

The surface area of the cube as a function of the length \(s\) of one side is \(6s^2\).

Step 1: Area of One Face

Calculate the area of one face of the cube by squaring the length of a side (s). So, the area of one face will be \(s^2\).

Step 2: Total Surface Area

Multiply the area of one face by 6 (the total number of faces on a cube) to find the total surface area. That would be \(6s^2\).

Key Concepts

Geometric Functions

In geometry, functions prove their value by describing relationships between different geometrical figures and measurements. A geometric function takes a geometric measurement as an input and gives another measurement as an output. Consider the simplicity and symmetry of a cube. Each side of this three-dimensional shape is equal in length, denoted as length s.

When you want to express the surface area of a cube as a function of the length of its side, you're creating a function where your input (the side length s) is directly related to your output (the surface area). This function allows us to generalize and quickly calculate the surface area for any cube as long as we know the value of one side.

Squaring a Length

Squaring a length is a fundamental mathematical operation that involves multiplying a number by itself. In the context of geometry, particularly with shapes like squares and cubes, squaring the length of a side gives you the area of one face of the object.

For example, if a cube has a side length of s, the area of one of its faces is s multiplied by s, or simply s squared (written as \(s^2\)). This squared term is a key element in understanding how areas and volumes are calculated, showing the power of exponentiation in geometric functions.

Total Surface Area Calculation

The total surface area of a cube can be calculated by considering the fact that it has six congruent faces—each being a square of equal area. Once you know the area of one face, as found by squaring the length of its side, you can find the total surface area by multiplying this amount by 6 (since there are 6 faces).

Mathematically, if one face has an area of \(s^2\), the total surface area (TSA) of the cube would be \(TSA = 6 \times s^2\). This formula encapsulates the entire calculation succinctly and effectively. It's an excellent demonstration of how a geometric function can simplify the process of calculating various properties of a three-dimensional object.