Problem 65 - Calculus with Concepts in Calculus

Question

Express the cost \(C\) of painting a rectangular wall as a function of the dimensions \(x\) and \(y\) (in meters) if the cost per square meter is \(\$ 0.30\).

Step-by-Step Solution

Answer

The cost function is \( C(x, y) = 0.30xy \).

Step 1: Understand the problem

We need to express the total cost of painting a wall as a function of its dimensions, where the wall is a rectangle and the cost per square meter is $0.30.

Step 2: Identify the area of the wall

The area of a rectangle is calculated by multiplying its length by its width. For a wall with dimensions \( x \) meters wide and \( y \) meters tall, the area \( A \) is given by \( A = x \times y \).

Step 3: Derive the cost function

The cost \( C \) of painting the wall depends on two factors: the area of the wall and the cost per square meter. Since the area is \( x \times y \) and the cost per square meter is $0.30, the total cost can be expressed as a function of \( x \) and \( y \): \( C(x, y) = 0.30 \times (x \times y) \).

Step 4: Simplify the cost function

Combine the terms to obtain the simplified function: \( C(x, y) = 0.30xy \). This is the cost function of the wall given its dimensions.

Key Concepts

Cost Function

A cost function in mathematics and economics represents the total cost of producing a certain quantity of goods or services. In the context of our exercise, the cost function specifically deals with calculating the total expense for painting a rectangular wall. Here, we take into account both the size of the wall and the price per square meter.

To derive the cost function, we consider that each square meter of the wall incurs a fixed cost of \( \$0.30 \). Therefore, the cost function is a direct product of the wall's area and this constant rate. The formula thus becomes \( C(x, y) = 0.30 \times (x \times y) \).

This formula neatly encapsulates how changes in the dimensions (width \( x \) and height \( y \)) will linearly affect the total cost \( C \). It's a simple yet powerful representation of how we can relate physical dimensions to economic expenditure. By understanding and deriving this, you can easily calculate costs for different wall sizes without having to start from scratch each time.

Area Calculation

The calculation of an area is a fundamental concept when managing tasks involving space, such as painting a wall, tiling a floor, or carpeting a room. The area gives you a measure of how much surface is covered. For a rectangular shape, the area is straightforward to compute.

A rectangle’s area is calculated as the product of its length and its width, typically expressed as \( A = l \times w \). In the specific exercise, the rectangle refers to a wall with a width of \( x \) meters and a height of \( y \) meters. Thus, the formula becomes \( A = x \times y \).

This essentially means that to find out how much surface you have (in this case, how much space needs painting), you'll multiply the two dimensions. This formula provides the foundation for other calculations, like determining costs, as it yields the total number of square meters you have.

Rectangular Geometry

Rectangular geometry is a branch of geometry focused on properties and relations involving rectangles, one of the most familiar shapes in geometry. Rectangles are quadrilaterals with opposite sides that are parallel and equal, featuring four right angles. They are prevalent in both natural and built environments.

Understanding the properties of rectangles is essential when dealing with projects involving such shapes. For instance, when calculating areas or perimeters, you rely on knowing that the opposite sides of a rectangle are not only parallel but equal in length.

In the problem at hand, the rectangular wall provides a practical example. Knowing the wall's dimensions allows us to apply these geometric principles to find the necessary area. This rectangle-based approach is often the starting point in building larger geometric and mathematical concepts, like forming cost functions or more complex architectural projects.