Problem 66 - 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 \(\$ 0.30\) and the wall contains a window 1 square meter in area.

Step-by-Step Solution

Answer

The cost function is given by \( C = 0.30(xy - 1) \).

Step 1: Calculate the Wall Area

The wall is rectangular with dimensions given by \( x \) and \( y \). Hence, the area of the wall can be calculated as \( A = x imes y \) square meters.

Step 2: Compute Window Area

The area of the window, which does not need to be painted, is given as 1 square meter.

Step 3: Determine Paintable Area

Subtract the window area from the total wall area to find the paintable area. Thus, the paintable area is \( x imes y - 1 \) square meters.

Step 4: Determine the Cost Function

The cost to paint each square meter is \$0.30. Therefore, multiply the paintable area by 0.30 to find the total cost. The cost function \( C \) is: \[ C = 0.30 imes (x imes y - 1) \]

Key Concepts

Area Calculation

When calculating the area of a surface, we often deal with simple geometric shapes like rectangles. For a rectangle, the area is determined by multiplying the length by the width. This gives us the total surface area expressed in square units, such as square meters.
In the context of a wall, if we let the length be denoted by \( x \) and the width by \( y \), the area is calculated as \( A = x \times y \). However, context can influence the area considered, especially if there are non-paintable sections, like windows.
Here, once the total area of the wall is found through this product, adjustments are made for any spaces, like windows, which cover 1 square meter. Thus, the actual area to consider for certain calculations would be this adjusted area.

Rectangular Wall

Rectangles are one of the most fundamental shapes in geometry. Walls are often rectangular, making them easy to work with in calculations. For any rectangular object, the dimensions are defined by two measurements: its length and width, noted here as \( x \) and \( y \).
Understanding how these dimensions apply to a rectangular wall helps in various calculations. Not only for area but also for determining material needs, such as paint. In scenarios like these, it's crucial to ensure all measurements are accounted for, including any features that might affect their uniformity, like windows or doors.
In our case, the window reduces the paintable area, reminding us that visualizing and listing all parts of a wall can guide accurate area and resource evaluations.

Cost Per Square Meter

The cost per square meter is a crucial value in many real-world applications. Knowing the price for each square meter helps estimate the full cost of any project, whether it involves flooring, tiling, or painting. This can guide budgeting and ensure efficient resource allocation.
In this exercise, the cost is given as \(0.30 per square meter. To find the total cost, you multiply the number of paintable square meters by \)0.30. Once non-paintable areas like windows are factored out, you have the adjusted area that determines the total expense.
The equation for total cost becomes: \( C = 0.30 \times (x \times y - 1) \). This formula provides developers, planners, or homeowners with an easy way to compute the cost associated with changes in dimensions or the cost per unit area.