Problem 26 - Prealgebra

Question

A model airplane is built with a wing span of 23 inches. The actual length of the wing is 92 feet. Find the scale factor. A. 1 inch \(=4\) feet B. 1 inch \(=8\) feet C. 1 inch \(=12\) feet D. 1 inch \(=16\) feet

Step-by-Step Solution

Answer

The scale factor is 1 inch = 4 feet (Option A).

Step 1: Understanding the Problem

We need to find the scale factor between the model airplane wing span and the actual wing span. The model's wing span is given in inches and the real airplane's wing span in feet. We look for the ratio of these spans to find the answer.

Step 2: Conversion to the Same Unit

Convert the actual wing span from feet to inches since both measurements need to be in the same unit for a proper comparison. As there are 12 inches in a foot, the actual wing span in inches is calculated as follows: \[ 92 \text{ feet} \times 12 \text{ inches/foot} = 1104 \text{ inches} \]

Step 3: Calculate the Scale Factor

With both spans now in inches, we can determine the scale factor. Divide the actual wing span in inches by the model wing span in inches to find how many actual inches correspond to one model inch: \[ \frac{1104 \text{ inches}}{23 \text{ inches}} \approx 48 \] Since this doesn't directly match any of the provided options, we realize that the scale in options must mean the feet equivalent of our step 2 and this is a key insight.

Step 4: Comparing Against Choices

Now we'll check which of the given options matches the calculated scale factor. Our conversion showed that \[ 1 \text{ model inch} = 48 \text{ inches of actual model} \]If we convert to feet, since each 1 model inch is 48 inches or 48 inches/12 (feet in inch), this gives:\[ \approx 4 \text{ feet} \] This calculation matches option A: 1 inch = 4 feet.

Key Concepts

Conversion Units

When solving problems like the scale factor of a model airplane, it's vital to work with common units, which are the same for all measurements involved. In the example exercise, one measurement is in inches, while the other is in feet. To allow for an accurate comparison, we must convert feet into inches. This means remembering the basic conversion rate: there are 12 inches in a foot.

• Conversion formula: Total inches = Total feet × 12
• For our example: 92 feet becomes 1104 inches

Understanding and using conversion units correctly is foundational for calculating scale factors accurately.

Ratio

A ratio is a way to compare two quantities, showing how many times one value contains or is contained within the other. In our problem-solving exercise, we need to find the ratio of the actual wing span to the model wing span.

• This involves aligning both measurements to the same unit first.
• After conversion, the actual span (1104 inches) and model span (23 inches) can be used to find a scale factor ratio.
• The ratio shows that every model inch corresponds to 48 actual inches.

Ratios help highlight the relationship between different-sized objects in a comparison, such as models and real-life counterparts.

Wing Span Measurement

Measuring the wing span requires accuracy, particularly when scaling down real-life measurements to create models. In this case, we are dealing with the span of an airplane wing, which typically involves larger numbers in real life than in models. The span is the distance from one wing tip to the other, an essential factor when designing models.

• The model’s wing span is 23 inches, whereas the real wing spans 92 feet.
• Properly translating and converting these measurements allows for accurate modeling and better understanding of scale factors.

Accurate measurements are the first step to solving scale challenges in model building.

Model Airplane Calculations

To accurately design and assess a scale model, mathematical calculations are necessary. These ensure that a model accurately represents its larger counterpart, maintaining the structural and aesthetic proportions.

• Starting with the actual dimensions, convert them appropriately (e.g., from feet to inches).
• Calculate the scale factor, which is how the model size compares to the actual size (in our case, the scale factor is 1:4).
• This factor can be used to derive or adapt other dimensions, ensuring full accuracy for the model build.

Such calculations ensure that the model not only looks correct but also maintains essential proportions for potential functional analyses or aesthetic fidelity.