4.1.1: Lots of Flags
Lesson
Let's explore the U.S. flag.
Exercise \(\PageIndex{1}\): Scaled or Not?
1. Which of the geometric objects are scaled versions of each other?
2. Pick two of the objects that are scaled copies and find the scale factor.
Exercise \(\PageIndex{2}\): Flags are Many Sizes
One standard size for the United States flag is 19 feet by 10 feet. On a flag of this size, the union (the blue rectangle in the top-left corner) is \(7\frac{5}{8}\) feet by \(5\frac{3}{8}\) 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.
- Share your answer with another group that used a different option. What do your dimensions have in common?
Exercise \(\PageIndex{3}\): What Percentage is the Union?
On a U.S. flag that is 19 feet by 10 feet, the union is \(7\frac{5}{8}\) feet by \(5\frac{3}{8}\) feet. For each question, first estimate the answer and then compute the actual percentage.
- What percentage of the flag is taken up by the union?
- What percentage of the flag is red? Be prepared to share your reasoning.
Are you ready for more?
The largest U.S. flag in the world is 225 feet by 505 feet.
- Is the ratio of the length to the width equivalent to \(1:1.9\), the ratio for official government flags?
- If a square yard of the flag weighs about 3.8 ounces, how much does the entire flag weigh in pounds?
Summary
Imagine you have a painting that is 15 feet wide and 5 feet high. To sketch a scaled copy of the painting, the ratio of the width and height of a scaled copy must be equivalent to \(15:5\). What is the height of a scaled copy that is 2 feet across?
| width | height |
|---|---|
| \(15\) | \(5\) |
| \(2\) | \(h\) |
We know that the height is \(\frac{1}{3}\) the width, so \(h=\frac{1}{3}\cdot 2\) or \(\frac{2}{3}\).
Sometimes ratios include fractions and decimals. We will be working with these kinds of ratios in the next few lessons.
Glossary Entries
Definition: Percentage
A percentage is a rate per 100.
For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.
Practice
Exercise \(\PageIndex{4}\)
A rectangle has a height to width ratio of \(3:4.5\). Give two examples of dimensions for rectangles that could be scaled versions of this rectangle.
Exercise \(\PageIndex{5}\)
One rectangle measures 2 units by 7 units. A second rectangle measures 11 units by 37 units. Are these two figures scaled versions of each other? If so, find the scale factor. If not, briefly explain why.
Exercise \(\PageIndex{6}\)
Ants have 6 legs. Elena and Andre write equations showing the proportional relationship between the number of ants, \(a\), to the number of ant legs \(l\). Elena writes \(a=6\cdot l\) and Andre writes \(l=\frac{1}{6}\cdot a\). Do you agree with either of the equations? Explain your reasoning.
(From Unit 2.2.2)
Exercise \(\PageIndex{7}\)
On the grid, draw a scaled copy of quadrilateral ABCD with a scale factor \(\frac{2}{3}\).
(From Unit 1.1.4)
Exercise \(\PageIndex{8}\)
Solve each equation mentally.
- \(\frac{5}{2}\cdot x=1\)
- \(x\cdot\frac{7}{3}=1\)
- \(1\div\frac{11}{2}=x\)
(From Unit 1.1.5)
Exercise \(\PageIndex{9}\)
Lin has a scale model of a modern train. The model is created at a scale of 1 to 48.
- The height of the model train is 102 millimeters. What is the actual height of the train in meters? Explain your reasoning.
- On the scale model, the distance between the wheels on the left and the wheels on the right is \(1\frac{1}{4}\) inches. The state of Wyoming has old railroad tracks that are 4.5 feet apart. Can the modern train travel on those tracks? Explain your reasoning.
(From Unit 1.2.5)