Problem 52 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. On a blueprint, a hallway is \(45.6 \mathrm{cm}\) long. The scale is \(1.2 \mathrm{cm}=1.0 \mathrm{m} .\) How long is the hallway?

Step-by-Step Solution

Answer

The hallway is 38 meters long.

Step 1: Understanding the Scale

The blueprint uses a scale where 1.2 cm represents 1.0 m in real life. Therefore, to find the actual length of any object from the blueprint, multiply the length on the blueprint by the conversion factor from cm to m.

Step 2: Setting Up the Conversion

Since each 1.2 cm on the blueprint represents 1.0 m in reality, we set up a proportion or use the conversion factor to determine the actual length:\[\text{Actual length} = \frac{1.0 \text{ m}}{1.2 \text{ cm}} \times \text{length on blueprint}\]

Step 3: Calculating the Real Length of the Hallway

The hallway measures 45.6 cm on the blueprint. So, we substitute this into our conversion formula:\[\text{Actual length} = \frac{1.0 \text{ m}}{1.2 \text{ cm}} \times 45.6 \text{ cm}\]

Step 4: Performing the Calculation

Calculate the actual length:\[\text{Actual length} = \frac{1.0 \text{ m}}{1.2} \times 45.6 = 38 \text{ m}\]

Step 5: Verifying the Result

We verify the result by considering the reasonableness: If 1.2 cm equals 1.0 m, then indeed, 45.6 cm should convert to a much larger measurement in meters, specifically 38 meters, as per our calculation.

Key Concepts

Blueprint Interpretation

Interpreting a blueprint is an essential skill in many fields, such as architecture and engineering. A blueprint is a detailed plan that represents various building components proportionately. To decode a blueprint accurately, one must understand the symbols, scale, and how measurements are translated into real-world dimensions.
Blueprint dimensions are not the actual sizes of structures; they are scaled versions.

• Each line and feature corresponds to a component in a building.
• The scale provided on the blueprint is vital for interpreting the drawings correctly.

Familiarizing yourself with the standard symbols and units used in the blueprint can make interpretation easier, enabling efficient communication and planning.

Proportions

Proportions are mathematical statements that show two ratios are equal. In the context of blueprints, proportions allow us to scale measurements accurately from a drawing to real life.
To set up a proportion, you use the scale from the blueprint. For example, if 1.2 cm is equal to 1.0 meter, 45.6 cm from the blueprint can be set up in the following proportion to find the real-world length:
\[ \frac{1.2 \text{ cm}}{1.0 \text{ m}} = \frac{45.6 \text{ cm}}{x \text{ m}} \]
Here, solving for \( x \) gives us the real-life measurement. Proportions maintain the relationship between different quantities, ensuring accurate conversion between scales.

Mathematical Scaling

Mathematical scaling is the process of resizing objects in a proportional manner. It involves adjusting the size of an object in a drawing according to a specified ratio or scale.

• When we say a drawing is at a certain scale, it means each unit of measure on the diagram is equal to a specific number of units in reality.
• In our example, the scale was given as \(1.2 \text{ cm} = 1.0 \text{ m}\).

To scale up from blueprint to real-world size, multiply the blueprint measurement by the conversion factor, or carry out a calculation using proportions. Scaling ensures that all building elements are proportionately sized, maintaining the integrity of the design in real dimensions.

Length Measurement

Length measurement on a blueprint requires the conversion of the measured length into real-life dimensions using the given scale. In the example provided, a length of 45.6 cm on the blueprint needs to be translated into meters.
The conversion involves multiplying the blueprint measurement by the scale conversion factor, as shown: \[ \text{Actual length} = \frac{1.0 \text{ m}}{1.2 \text{ cm}} \times 45.6 \text{ cm} \]
This calculation results in a real-world hallway length of 38 meters.

• Accurate length measurement is crucial for constructing buildings as per design specifications.
• Misinterpretation can lead to significant errors in construction and cost.

Measuring lengths correctly ensures the final product matches the intended design.