Problem 7 - Prealgebra

Question

On a set of architectural drawings for a new school, the scale is \(\frac{1}{2}\) inch \(=\) 9 feet. Find the actual length of each room. $$\begin{array}{|l|c|}\hline \quad \quad \quad \text { Room } & \text { Drawing Distance } \\ \hline \text { Nurse's Office } & 2 \text { inches } \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The actual length of the Nurse's Office is 36 feet.

Step 1: Understand the Scale

The scale provided on the drawing is \(\frac{1}{2}\) inch = 9 feet. This scale helps us understand the real-world distance in terms of the distance shown on the drawings.

Step 2: Set Up a Proportion

To find the actual length of the room, we need to set up a proportion based on the provided scale. We have \(\frac{1}{2}\) inch equivalent to 9 feet, and the drawing shows 2 inches. Therefore, the proportion is \( \frac{1}{2} : 9 = 2 : x \), where \( x \) is the actual length in feet.

Step 3: Solve the Proportion

Cross-multiply to solve the proportion: \( \frac{1}{2} \times x = 9 \times 2 \). This equation becomes \( \frac{x}{2} = 18 \). Multiply both sides by 2 to find \( x \).

Step 4: Calculate the Actual Length

By multiplying both sides of the equation \( \frac{x}{2} = 18 \) by 2, we get \( x = 36 \). This means the actual length of the Nurse's Office is 36 feet.

Key Concepts

Scale Drawing

Scale drawings are simplified representations of objects or spaces, where each measurement in the drawing corresponds proportionally to a measurement in the real world. Think of it as using a map to navigate your route - the map itself is much smaller than the actual landscape, but it's crafted in a way that accurately represents distances.

• Scale often appears as a ratio, like in our problem: \( \frac{1}{2} \) inch = 9 feet.
• Each unit on the drawing corresponds to the actual unit size, allowing us to infer real-world dimensions from the drawing directly.
• This is crucial in fields like architecture, engineering, or any design work where precise measurements must translate to actual sizes accurately.

Understanding how scale drawings work helps in visualizing a space before it exists and in ensuring that the design fits the intended context. It allows architects and designers to convey detailed plans effectively while minimizing physical space needed for the sketch itself.

Measurement Conversion

Measurement conversion in scale drawings involves translating a length from the drawing to its real-world size. This process is critical in maintaining accuracy as dimensions shift from theoretical plans to practical applications. For instance, in the given problem, the scale is set to \( \frac{1}{2} \) inch equating to 9 feet.
When you're converting:

• Determine from the scale how much of the real-world area one unit of the drawing covers.
• Use this relationship to find actual dimensions by setting up a proportion. - Example: If 2 inches exist on the drawing, with our scale, we can design a proportion. - Solve this proportion to find the real-world length.
• Apply mathematical operations carefully to ensure each conversion step is accurate.

Conversion ensures that designs are realizable and resources are adequately accounted for. This helps in preventing errors during construction and assists in planning for material needs effectively.

Mathematical Reasoning

Mathematical reasoning is the process of thinking logically and systematically when approaching problem-solving. In the exercise example, it guides us in setting up equations and using proportions to solve for unknowns. This pathway involves understanding how each variable and data point interconnects within the problem.
Key elements include:

• Understanding Relationships: Recognize how each part of the proportion relates and influences the other. Understanding that \( \frac{1}{2} \) inch relates to 9 feet is essential before solving the problem.
• Proportional Thinking: Use cross-multiplication when dealing with ratios to find the value of the unknown variable.
• Step-by-Step Calculation: Follow a logical process to solve equations, particularly when unknowns are involved. This means maintaining equality by applying operations uniformly across both sides of an equation.

This approach helps cultivate skills for tackling complex problems and is applicable beyond math, supporting reasoning ability in daily decisions and scientific inquiries, too.