Marissa's Patio Painting Project Calculating Paintable Area

Hey guys! Let's dive into a super practical math problem today. Imagine Marissa, who's decided to give her rectangular patio a fresh coat of paint. Sounds like a fun weekend project, right? But here's the thing: she's got this awesome bench sitting on the patio that she doesn't want to paint. So, how do we figure out exactly how much of the patio she does need to paint? That's where a little bit of math comes in handy!

Understanding the Problem

Before we jump into equations, let's break down what we know. We're dealing with a rectangular patio, which is key because we know how to calculate the area of a rectangle. The area of a rectangle is simply its length multiplied by its width. But, Marissa isn't painting the whole patio. There's this bench taking up some space. Think of it like this: we need to find the total area of the patio and then subtract the area of the bench. This will give us the paintable area. To effectively calculate the paintable area, understanding the dimensions of both the patio and the bench is crucial. The patio's length and width will determine the total area, while the bench's dimensions will tell us how much area needs to be subtracted. Visualizing this as a rectangle with a smaller rectangle (the bench) inside it can be helpful. Remember, the goal here is to find the area that Marissa will actually be painting, which means accounting for the unpainted bench. This involves a straightforward application of geometric principles and a little bit of algebraic thinking to represent the situation mathematically. So, let’s get our thinking caps on and figure out how to set up the equation!

Defining Variables and Setting Up the Equation

Okay, so let's get down to the nitty-gritty. To write an equation, we need to use variables. Variables are just like stand-ins for numbers we might not know yet. In this case, we need variables for the length and width of the patio, and probably for the length and width of the bench too. Let's say:

• L = Length of the patio
• W = Width of the patio
• l = Length of the bench
• w = Width of the bench

Remember, the area of a rectangle is length times width. So, the total area of the patio is L * W. Now, the area of the bench (the part Marissa isn't painting) is l * w. To find the area Marissa will be painting (A), we need to subtract the bench's area from the patio's total area. This is where our equation comes in! The equation to determine the area, A, of the patio that will be painted is:

A = (L * W) - (l * w)

See? It's not as scary as it might have seemed at first! This equation basically says: the paintable area (A) is equal to the total patio area (L * W) minus the bench area (l * w). This equation is a powerful tool for solving the problem, allowing us to plug in specific measurements and calculate the exact area Marissa needs to cover with paint. Furthermore, understanding the logic behind the equation is just as important as the equation itself. It helps to visualize the scenario and ensure that each part of the equation corresponds to a real-world element of the problem. For instance, L * W represents the entirety of the patio surface, while l * w represents the portion that will remain untouched. By subtracting the latter from the former, we isolate the area that requires painting. This method of thinking can be applied to various similar problems, making it a valuable skill to develop. So, with our variables defined and our equation set, we are well-equipped to tackle any specific measurements Marissa's patio and bench might have.

Applying the Equation: An Example

Let's make this even clearer with an example. Let's say Marissa's patio is 10 feet long and 8 feet wide. So, L = 10 feet and W = 8 feet. And let's say her bench is 4 feet long and 2 feet wide, making l = 4 feet and w = 2 feet. Now we can plug these values into our equation:

A = (L * W) - (l * w) A = (10 * 8) - (4 * 2) A = 80 - 8 A = 72

So, the area Marissa will be painting is 72 square feet. Pretty cool, huh? This example beautifully illustrates the practicality of the equation we have developed. By assigning concrete values to the variables, we can see the equation in action, transforming abstract symbols into a tangible result. The calculation itself is straightforward, involving simple multiplication and subtraction, but the underlying concept is powerful. It shows how mathematical formulas can be used to solve real-world problems, from home improvement projects to more complex engineering tasks. Moreover, this example reinforces the importance of units. Since we measured the lengths in feet, the area is naturally expressed in square feet. This is a crucial detail to remember in any calculation involving physical quantities, as incorrect units can lead to significant errors. So, by walking through this example, we not only gain confidence in using the equation but also deepen our understanding of its significance and applicability.

Why This Equation Matters

You might be thinking,