High School

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1.2: Flags Are Many Sizes

One standard size for the United States flag is 9 feet by 10 feet. On a flag of this size, the union (the blue rectangle in the top-left corner) measures [tex]7 \frac{5}{8}[/tex] feet by [tex]5 \frac{3}{8}[/tex] feet.

There are many places that display flags of different sizes:
- Many classrooms display a U.S. flag.
- Flags are often displayed on stamps.
- There was a flag on the space shuttle.
- Astronauts on the Apollo missions had a flag on a shoulder patch.

Choose one of the four options and decide on a size that would be appropriate for this flag. Find the size of the union.

Answer :

Of course! Let's work through this problem step by step.

We are looking for an appropriate size for a flag and the corresponding size of its union, given different contexts such as in a classroom. For this example, we'll find appropriate dimensions for a classroom flag.

### Standard Flag Sizes:
- The standard U.S. flag is 9 feet by 10 feet.
- The union (blue rectangle) of this flag is [tex]\(7 \frac{5}{8}\)[/tex] feet by [tex]\(5 \frac{3}{8}\)[/tex] feet.

### Classroom Flag Size:
Let's assume a suitable size for a classroom flag is 3 feet by 5 feet. We need to determine the size of the union for this reduced-size flag. To do this, we'll use a scaling factor based on the smaller flag size.

### Calculating the Union Size for the Classroom Flag:

1. Determine the Scaling Factor:
- The scaling factor is derived from the ratio of the new flag's length to the standard flag's length.
- [tex]\(\text{Scaling Factor} = \frac{\text{Classroom Flag Length}}{\text{Standard Flag Length}}\)[/tex]
- Which is [tex]\(\text{Scaling Factor} = \frac{3}{9} = \frac{1}{3}\)[/tex].

2. Calculate the Union's Length for the Classroom Flag:
- Multiply the union's standard length by the scaling factor.
- Standard union length = [tex]\(7 \frac{5}{8}\)[/tex] feet which is [tex]\(7.625\)[/tex] feet.
- Classroom union length = [tex]\(7.625 \times \frac{1}{3} = 2.5417\)[/tex] feet.

3. Calculate the Union's Width for the Classroom Flag:
- Multiply the union's standard width by the scaling factor.
- Standard union width = [tex]\(5 \frac{3}{8}\)[/tex] feet which is [tex]\(5.375\)[/tex] feet.
- Classroom union width = [tex]\(5.375 \times \frac{1}{3} = 1.7917\)[/tex] feet.

Thus, for a classroom flag that is 3 feet by 5 feet, the dimensions of the union would be approximately 2.54 feet by 1.79 feet. These are the dimensions that maintain the same proportions as the standard-sized flag.

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