Problem 36 - Prealgebra

Question

DECORATING A gallon of paint covers about 350 square feet. A painter estimates the area to paint by multiplying the combined wall lengths by the height and subtracting 15 square feet for each window or door. Suppose a rectangular room measures 15 feet long by 12 feet wide. The room is 9 feet high and has two windows and two doors. How many gallons of paint are needed to paint the room using two coats of paint?

Step-by-Step Solution

Answer

3 gallons of paint are needed.

Step 1: Calculate the Perimeter of the Room

First, we need to calculate the total length of the walls around the room. The perimeter of a rectangular room is calculated by adding twice the length and twice the width. Thus,\[\text{Perimeter} = 2 \times (15\text{ ft} + 12\text{ ft}) = 2 \times 27\text{ ft} = 54\text{ ft}.\]

Step 2: Calculate the Total Wall Area

Multiply the perimeter by the height of the walls to get the total area of the walls in the room.\[\text{Total Wall Area} = 54\text{ ft} \times 9\text{ ft} = 486\text{ sq ft}.\]

Step 3: Subtract the Area for Windows and Doors

The room has two windows and two doors. Each window and door subtracts 15 square feet from the total wall area.\[\text{Total Area for Windows and Doors} = 4 \times 15\text{ sq ft} = 60\text{ sq ft}. \]Subtract this from the total wall area:\[\text{Effective Area to Paint} = 486\text{ sq ft} - 60\text{ sq ft} = 426\text{ sq ft}.\]

Step 4: Calculate the Area for Two Coats of Paint

Since the room requires two coats of paint, multiply the effective area by 2.\[\text{Area for Two Coats} = 2 \times 426\text{ sq ft} = 852\text{ sq ft}.\]

Step 5: Calculate the Number of Gallons Needed

A single gallon of paint covers 350 square feet. Divide the total painting area needed by the coverage of one gallon to determine how many gallons are required.\[\text{Gallons Needed} = \frac{852\text{ sq ft}}{350\text{ sq ft/gallon}} \approx 2.43.\]Since you can only buy whole gallons, round up to the nearest whole number. Therefore, you need 3 gallons of paint.

Key Concepts

Area Calculation

Let’s dive into the idea of area calculation. When we talk about the area in the context of painting a room, we are looking at the total surface that paint has to cover.
For walls, we first combine the lengths of all four sides of the room (known as the perimeter) and multiply by the height of the walls.
This gives us the total wall area.

• Use the formula: Total Wall Area = Perimeter × Height.
• Subtract the areas of windows and doors, as paint will not be applied there.

By understanding area calculation, you get the accurate surface dimension that needs paint, helping you estimate the resources you'll need.

Perimeter

The perimeter is a fundamental concept, especially when calculating paint for a room. It refers to the total length around any shape. To calculate it for a rectangular room:

• Add twice the length and twice the width together.
• For example, for a room measuring 15 feet long and 12 feet wide, the perimeter is 2 × (15 + 12) = 54 feet.

Knowing the perimeter helps in determining the entire stretch of walls that require painting. It's a primary step in measuring how much paint one needs.

Multiplication

Multiplication is a handy mathematical operation you'll see at almost every step when dealing with painting calculations.
It helps us scale measurements to their corresponding values, such as in determining the total wall area.

• When you multiply perimeter by height, you get the total wall area.
• Similarly, multiplying the area that paint should cover by the number of coats gives you the total area to be painted.

Using multiplication effectively means fewer mistakes when planning out your paint job.

Subtraction

Subtraction is key when we need to exclude certain measurements that don't require painting.
In our exercise, this means removing the areas taken up by windows and doors.

• Subtract the total area occupied by windows and doors from the total wall area.
• Each window and door accounts for 15 square feet. Multiply this by the number of windows and doors, and subtract from the overall wall area.

This operation ensures the paint covers only the necessary surfaces.

Geometry

Geometry involves understanding the properties and measurements of shapes.
With rooms being effectively 3D shapes, geometry helps in understanding how we wrap paint around the structure.

• Rooms are typically rectangular prisms, so knowing how to calculate for rectangles is important.
• Geometry helps estimate areas to cover and underscores why we measure parameters like perimeter and height.

Mastering geometric principles can make home painting projects more efficient and precise.