Question

The floor plan of a certain building has a scale of \(\frac{1}{4}\) in. \(=1 \mathrm{ft}\) and shows a room having an area of 40 in. \(^{2}\). What is the actual room area in square feet?

Short answer

The actual room area is \(40 \times 48 \text{in.}² \div 144 \text{in.}²/\text{ft.}² = 13,333.33 \text{ ft.}²\).

Step-by-step solution

Understand the Scale

First, understand that the scale \(\frac{1}{4}\text{ in.} = 1 \text{ft}\) means that every quarter inch on the floor plan corresponds to 1 foot in the actual room dimensions.

Convert the Scale to the Same Units

Since there are 12 inches in a foot, the scale can be rewritten in inches as \(\frac{1}{4}\text{ in.} = 12 \text{in.}\). Thus, every 1 inch on the plan equals to 48 inches (4 feet) in real life.

Calculate the Conversion Factor

To get the conversion factor for area, square the linear conversion factor. This means \(1 \text{ in.}² \text{ on the plan equals } 48 \text{ in.}² \text{ in real life (= } 4 \text{ ft.} × 12 \text{ in./ft.} = 48 \text{ in.})\).

Calculate the Actual Area

Multiply the area shown on the plan by the conversion factor. The actual area in square inches is \(40 \text{ in.}²\times 48 \text{ in.}² \text{ per in.}²\). Convert the result to square feet by dividing by \(12 \text{ in.}\times 12 \text{ in.}\).

Key concepts

Mathematical Scale Factor

In scale conversion, the mathematical scale factor is a critical aspect to comprehend. What it refers to is the ratio of a measurement on a scale model or drawing to the actual measurement in the real world.

For example, if a floor plan of a building designates that \(\frac{1}{4}\) inch is equivalent to 1 foot in reality, then the mathematical scale factor between the dimensions on the plan and the actual dimensions is 48 (since 1 foot is 12 inches, we multiply \(\frac{1}{4}\) by 48 to determine the equivalent number of inches in real-world measurement).

This factor is what allows us to translate measurements from the plan to the actual sizes, ensuring that everything we build or measure is accurate to the design. With the proper scale factor, we can scale up or down between models/drawings and reality efficiently and accurately.

Area Calculation

When it comes to measuring space, area calculation is fundamental. It leads to understanding how much of a two-dimensional flat surface is covered.

Calculating area is usually straightforward when dealing with regular shapes, such as rectangles or squares, where you often simply multiply length by width. However, when working with scaled drawings, it becomes a little trickier, as we need to apply the square of the linear scale factor to convert the area on a scaled drawing to the actual area.

In our exercise, the area of 40 square inches on the model needs to be scaled up by squaring our linear scale factor. This is because we're not only scaling a single dimension but two - both the length and the width of the space.

Unit Conversion

Unit conversion is a crucial skill across various scientific and engineering disciplines, and it's equally significant in everyday situations. It allows us to convert measurements from one system or scale to another — for instance, from inches to feet, or meters to kilometers.

In the context of our exercise, we first converted the scale from \(\frac{1}{4}\) inch to feet and then understood it as an equivalent scale in inches. We then used this to convert an area from square inches to square feet. Proper unit conversion is fundamental to ensure that area calculations reflect real-world sizes correctly and to enable clear communication of measurements.

Architectural Scale

Architectural scale is an essential concept in the design and construction industries. It is a specialized type of ratio used on drawings to represent an object's or space's actual size. Architects and builders rely on scales to create and read plans that would be otherwise impractical if drawn to full size.

In our exercise, an architectural scale of \(\frac{1}{4}\) inch equals 1 foot means that every inch measured in the drawing would need to be 'transformed' into the real-world measurement of 48 inches (which is 4 feet) to understand the actual size. This scale enables architects to represent complex structures on paper in a manner that is simpler to handle and communicate while maintaining accuracy in dimensions.