Problem 12 On a set of architectural drawin... [FREE SOLUTION]

Chapter 6: Problem 12

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. $$\begin{array}{|l|c|}\hline \quad \quad \quad \text { Room } & \text { Drawing Distance } \\ \hline \text { Exercise Room } & 3 \frac{3}{4} \text { inches } \\\\\hline\end{array}$$

Short Answer

Expert verified

The actual length of the Exercise Room is 67.5 feet.

Step by step solution

Understand the Scale

The problem states that $ \frac{1}{2} $ inch on the drawing equals 9 feet in reality. This means for every $ \frac{1}{2} $ inch on the drawing, the actual length is 9 feet.

Convert Mixed Number to Improper Fraction

The drawing distance given for the Exercise Room is $ 3 \frac{3}{4} $ inches. First, convert this mixed number to an improper fraction.\[3 \frac{3}{4} = \frac{15}{4} \text{ inches}\]

Calculate Actual Length

Use the scale to find the actual length by setting up a proportion: $ \frac{1}{2} $ inch corresponds to 9 feet. Therefore, $ \frac{15}{4} $ inches will correspond to $ x $ feet:\[\frac{1}{2} : 9 = \frac{15}{4} : x\] Solve for $ x $ by cross-multiplying and simplifying: \[\left( \frac{1}{2} \right) x = 9 \times \frac{15}{4} \\frac{x}{2} = \frac{135}{4} \x = \frac{135}{4} \times 2\x = \frac{270}{4} = 67.5\]

Interpret the Result

The calculated value means the actual length of the Exercise Room is 67.5 feet according to the scale.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Proportion

A proportion is a statement that two ratios are equal. It acts as a mathematical bridge that helps us understand relationships between different quantities. In our exercise, we use the given scale to create a proportion. Each part of the scale represents a real world distance compared to a distance on a map or drawing. This is essential in fields such as architecture and engineering, where you might want to know the real-life size based on a model or plan. To set up a proportion, follow these steps:

Identify the relationship or scale. For example, in the exercise: 1/2 inch equals 9 feet.
Convert the drawing measurement to something comparable. This might require changing a mixed number to an improper fraction, as is necessary in this problem.
Finally, solve for the unknown by cross-multiplying and simplifying the expression, which helps keep the relationship balanced.

Using proportions lets us extrapolate or determine any missing value when one side of a ratio and a relation are known.

Converting Mixed Numbers to Improper Fractions

When you're dealing with mixed numbers in complex problems, it's often easier to convert them into improper fractions. A mixed number is composed of a whole number and a fraction, making calculations cumbersome.Here's how to convert a mixed number into an improper fraction:

Multiply the whole number by the denominator of the fraction part.
Add this product to the numerator of the fraction part.
The total becomes the new numerator, while the denominator remains the same.

For instance, to convert the mixed number $3 \frac{3}{4}$ into an improper fraction:

Multiply the whole number 3 by the denominator 4, which gives 12.
Add the numerator 3 to that product, reaching a sum of 15.
This sum, 15, becomes the new numerator, over the original denominator, 4, to form $\frac{15}{4}$.

This conversion makes subsequent calculations, like setting up a proportion, much more straightforward.

Cross-Multiplying in Proportions

Cross-multiplying is a useful technique in solving proportions. By cross-multiplying, you eliminate the fractions, which makes it easier to solve for the unknown variable.To cross-multiply, follow these straightforward steps:

Set up your proportion, with two fractions set equal to each other.
Multiply the numerator of the first fraction by the denominator of the second fraction.
Then, multiply the denominator of the first fraction by the numerator of the second fraction.

Take, for example, the proportion from the exercise:\[ \frac{1}{2} : 9 = \frac{15}{4} : x\]Cross-multiply to solve for $x$:

Multiply $\frac{1}{2} \times x$, which is $\frac{x}{2}$.
Then multiply $9 \times \frac{15}{4}$ to get $\frac{135}{4}$.
Finally, isolate $x$ by multiplying $\frac{135}{4}$ by 2, resulting in $x = \frac{270}{4} = 67.5$.

This process cuts through the complexity, allowing you to find the solution with clarity.

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91影视

Short Answer

Step by step solution

Understand the Scale

Convert Mixed Number to Improper Fraction

Calculate Actual Length

Interpret the Result

Key Concepts

Understanding Proportion

Converting Mixed Numbers to Improper Fractions

Cross-Multiplying in Proportions

One App. One Place for Learning.

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