Problem 3 - Calculus: Graphical, Numerical, Algebraic

Question

the surface area \(S\) of a cube as a function of the length of the cube's edge \(e\) ; the surface area of a cube of edge length 5 \(\mathrm{ft}\)

Step-by-Step Solution

Answer

The surface area of a cube with an edge length of 5 ft is 150 square ft.

Step 1: Understand the formula

The surface area \(S\) of a cube as a function of the length of the cube's edge \(e\) is given by the formula \(S = 6e^2\). It's called so because a cube has 6 equal faces, and the area of each face is \(e^2\). Therefore, the total surface area is 6 times the area of one face.

Step 2: Substitute the value into the formula

Now replace \(e\) in the formula with 5 ft, \(S = 6 \cdot (5^2)\), to get the surface area for the cube of edge length 5 ft.

Step 3: Perform the calculations

Perform the calculation \(S = 6 \cdot 25\), which simplifies to \(S = 150\).

Key Concepts

Geometry in Cubes

Geometry is a branch of mathematics that deals with the properties and relationships of shapes in space. One of the most basic geometric shapes is the cube. A cube, often called a hexahedron, is a three-dimensional shape with six equal square faces. Understanding the geometry of a cube helps in calculating its surface area.

Each face of the cube is identical in shape and size. Since each face is a square, and all squares in a cube are congruent, the calculations involving its surface area become relatively straightforward. Geometry principles help us know that the sum of the areas of these faces gives us the total surface area. This is crucial when dealing with geometrical calculations of different three-dimensional shapes.

Role of Mathematical Functions

Mathematical functions allow us to express relationships between different quantities. For a cube, the relationship between its edge length and surface area can be expressed in function form.

The function is given by the equation \(S = 6e^2\). Here, \(S\) represents the surface area, while \(e\) is the edge length of the cube.

• The equation \(S = 6e^2\) expresses a direct correlation, where the surface area increases as the square of the edge's length.
• This function simplifies calculations, as knowing the edge length allows for an immediate computation of the cube’s surface area.

This function and others help in various fields of mathematics and science to perform efficient calculations.

Cube Calculation Simplified

Calculating the surface area of a cube is a straightforward process once you understand the formula involved. The cube calculation involves just one critical measurement — the edge length.

Here’s a brief guide to compute the surface area of a cube:

• Start by identifying the length of the cube's edge, denoted as \(e\).
• Use the formula for surface area: \(S = 6e^2\), where \(6\) accounts for the six faces of the cube.
• Substitute the given edge length into this formula. In our example, substitute \(e = 5\) ft, yielding \(S = 6 \times 5^2\).
• Compute the exponent first, \(5^2 = 25\), and then multiply by 6 to get the surface area: \(S = 150\, \text{square feet}\).

This method ensures an accurate calculation of the cube's surface area, efficiently connecting the geometrical properties and mathematical functions in use.