Problem 76 - Intermediate Algebra

Question

Drafting. In a scale drawing, a 280 -foot antenna tower is drawn \(7 \frac{1}{2}\) inches high. The building next to it is drawn \(2 \frac{1}{4}\) inches high. How tall is the actual building?

Step-by-Step Solution

Answer

The actual building is approximately 84 feet tall.

Step 1: Identify the Scale

To find the scale used in the drawing, compare the actual height of the antenna tower, 280 feet, to the drawn height of the tower, which is \(7\frac{1}{2}\) inches. Express this relationship as a scale ratio: the actual height (280 feet) is to the drawing height (7.5 inches).

Step 2: Calculate the Scale Ratio

Convert the improper fraction \(7\frac{1}{2}\) to a decimal, which is 7.5. Now, find the scale ratio using the expression:\[\text{Scale Ratio} = \frac{280 \text{ feet}}{7.5 \text{ inches}}\]Calculate the scale ratio:\[\text{Scale Ratio} = \frac{280}{7.5} \approx 37.33 \text{ feet per inch}\]

Step 3: Determine Building Height in Drawing

The height of the building in the drawing is given as \(2\frac{1}{4}\) inches. Convert this to a decimal number: \(2\frac{1}{4} = 2.25\) inches.

Step 4: Calculate Actual Height of the Building

Use the scale ratio to find the actual height of the building. Multiply the drawn height by the scale ratio:\[\text{Actual Height} = 2.25 \text{ inches} \times 37.33 \text{ feet per inch}\]Calculate the product:\[\text{Actual Height} \approx 83.9925 \text{ feet}\]

Step 5: Conclusion

Round the calculated height to a reasonable degree of precision, depending on the context. Here, we will round to two decimal places: \[\text{Actual Building Height} \approx 84.00 \text{ feet}\]

Key Concepts

Understanding Scale Ratios

When working with scale drawings, understanding the concept of a "scale ratio" is essential. The scale ratio is a relationship between the actual size of an object and the size of its representation in a drawing. This ratio simplifies complex measurements to manageable sizes in a drawing.
In the given exercise, the scale ratio is established by comparing the real height of a 280-foot antenna tower to its illustration on paper, which is 7.5 inches tall. The fundamental idea here is that one measurement represents the same multiple of another measurement consistently. So, for every inch in the drawing, there are a specific number of feet in reality.

• To calculate it, the actual measurement is divided by the drawing measurement.
• This calculation gives you a scale ratio, which in our case is approximately 37.33 feet per inch.

Working with scale ratios helps maintain proportionality and ensures that drawings accurately reflect real-world dimensions.

Proportional Reasoning and Its Applications

Proportional reasoning is the mathematical foundation that helps solve problems involving proportions or ratios. In our exercise, you're dealing with proportions to determine the actual size of something when given its scaled version on paper.
The understanding of proportion is pivotal. If the tower is represented with a factor of 37.33 feet for every inch in the drawing, this factor is consistently applied to all items in the same scaled environment. Therefore, this factor allows us to extrapolate actual sizes from their scale versions smoothly.

• Keep proportions consistent by maintaining the same scale factor across similar calculations.
• Apply proportional reasoning to deduce unknowns from the known informations using ratios.

By utilizing these proportional relationships, you can effectively translate measurements from one form to another, ensuring accurate communication of complex real-world dimensions into more digestible representations.

Mastering Measurement Conversion

Measurement conversion plays a key role in resolving problems found in scale drawings or other contexts involving scales and ratios. It involves translating a measurement from one unit to another, ensuring the numerical relationship between different scales or systems is consistent.
In this exercise, you've converted inches to feet by applying a scale ratio (such as translating 2.25 inches on paper to 84 feet in actuality). This entails multiplying the smaller measurement by the scale to get the equivalent real-world size.

• Begin by ensuring all measurements are in compatible units before applying the ratio.
• Convert fractions to decimals for easier multiplication and calculation consistency.
• Use calculators for precise measurement conversion, especially in contexts requiring decimal calculations.

Understanding measurement conversions ensures you can accurately translate model or draft sizes to their full-size counterparts, crucial for accurate designs and constructions.