Problem 326 - Calculus Volume 1

Question

For the following exercises, consider a closed rectangular box with a square base with side \(x\) and height \(y\) Find an equation for the surface area of the rectangular box, \(S(x, y) .\)

Step-by-Step Solution

Answer

The equation for surface area is \(S(x, y) = 2x^2 + 4xy\).

Step 1: Understand the Shape

The figure described is a rectangular box with a square base. The sides of the base are both of length \(x\) and the height of the box is \(y\). We need to express the surface area of this box.

Step 2: Analyze Parts of the Surface Area

The box has two types of surfaces: the square base and top, and the four rectangular sides. The base and top each have an area of \(x^2\), and there are two such surfaces. The four sides are rectangles with areas \(x \times y\), with four such sides.

Step 3: Calculate Total Surface Area

To find the total surface area \(S(x, y)\), sum the areas of all the surfaces: The two square areas contribute \(2x^2\), and the four rectangular areas contribute \(4xy\). Thus, the total surface area is given by:\[S(x, y) = 2x^2 + 4xy\]

Step 4: Equation for Surface Area

The surface area of the closed rectangular box with a square base and height \(y\) is expressed as:\[S(x, y) = 2x^2 + 4xy\]. This equation accounts for all the faces of the box.

Key Concepts

Square Base

When we talk about the square base of a rectangular box, it means that the bottom part of the box is in the shape of a perfect square. This is important because a square has all its sides of equal length. In our exercise, the length of each side of the square base is denoted by \(x\). Since it's a square, the area of this base is calculated by multiplying one side by itself, which is \(x \times x = x^2\).

This concept is vital because the square base is not just at the bottom of the box, but the box has another identical square surface at the top. Therefore, when calculating the total area contributed by the square sections, we consider both the top and the bottom surfaces.

• Both the top and bottom are squares with area \(x^2\).
• Since there are two of these surfaces, their combined area adds up to \(2x^2\).

Understanding the square base helps in simplifying the surface area calculations by quickly identifying these two identical large surfaces of the box.

Rectangular Sides

Apart from the square base and the top, the box has four other surfaces known as the rectangular sides. Each of these sides is a rectangle, and the dimensions are determined by the length of the base \(x\) and the height of the box \(y\). The area for any rectangle is calculated by multiplying its length by its width. Hence, the area of each rectangular side is given by \(x \times y\).

Since the box has a total of four such rectangular sides, we need to consider all of them in the surface area calculation.

• Each rectangular side contributes an area of \(x \times y\).
• With four rectangular sides, their combined area becomes \(4xy\).

This component is critical in determining the total surface area as it forms the major peripheral part of the box.

Surface Area Calculation

The surface area of any three-dimensional object, like our rectangular box, is the total area exposed to the outside. It's crucial because it helps understand how much material wraps around or covers the object.

In our rectangular box with a square base and height \(y\), calculating the surface area involves combining the areas of both the square and rectangular portions. As laid out in the solution:

• The two square surfaces contribute \(2x^2\).
• The four rectangular sides add up to \(4xy\).

The formula for the surface area of this box becomes:\[S(x, y) = 2x^2 + 4xy\]
This equation succinctly combines all the contributing areas. It serves as an efficient way to determine the surface needed to enclose such a box entirely. Utilizing this formula, you can easily compute how much material is required for any given \(x\) and \(y\), making it very practical in real-world applications such as packaging and construction.