Another Math Problem

In our last What's Up In Space posting we challenged you to measure the moon from earth. That is possible because we can use math to help us figure out how to do it.

We have another math challenge for you this week. The problems below were sent to us from Marisa Diaz Mendez, a middle-school math teacher. See if you can figure out the answers. Please email us your answers, and we will mail you a free Space Science Group t-shirt for participating. In your email please include your grade, your school, your teacher, and your t-shirt size. Use this email link: ssg@nsula.edu.

And, here's the math challenge. Good luck!

After the recent events in the United States of America, flags can be seen everywhere you look! For many people, finding flags to buy in stores is next to impossible due to the great demand. Therefore, many of us got creative, and started making our own handmade flags out of paper, pins, construction paper, fabric, etc. Ms. Mendez decided to have her classes look into the proper dimensions of the United States flag and she came up with some great math problems!

Her classes found the following information about the official dimensions of the US flag (fixed by an executive order signed by Pres. Taft on Oct 29, 1912):

When flying horizontally, the overall width (top to bottom) and length (left to right) is in the ratio of 1:1.9. There are 13 stripes of equal width. The width (top to bottom) of the blue field containing the stars is 7/13 of the overall width of the flag; that is, from the top of the flag to the bottom of the 7th stripe. The length (left to right) of blue field is .76 of the overall width of the flag.

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As Big As Possible: Ana wants to draw a US flag on a standard 8 by 11-inch sheet of construction paper. What are the dimensions, in inches, of the largest flag she can draw if she approximates the ratio of the width:length as 1:2?

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Red, White, and BLUE: Antonio is making a US flag with 2-inch wide stripes. If his flag has the correct ratios, what is the area of the entire rectangular blue field (including the stars)?

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As American As Apple "Pi": Our current flag has 50 stars, arranged in nine alternating rows of six and five stars. The original flag had 13 stars arranged in a circle, such that the outermost tip of each of the 13 stars was on the circumference of the circle. If the distance between the outermost tips of consecutive stars is 4 inches on the circumference of the circle, what is the radius of the circle, in inches? Express your answer in terms of pi.

Thank you to Marisa Diaz Mendez for sharing this great problem set!

Credit: Adapted from NASA's MATHCOUNTS

ANSWERS TO THE MATH CHALLENGE

Here are the answers to last week's math challenge.

These were the parameters:
When flying horizontally, the overall width (top to bottom) and length (left to right) is in the ratio of 1:1.9. There are 13 stripes of equal width. The width (top to bottom) of the blue field containing the stars is 7/13 of the overall width of the flag; that is, from the top of the flag to the bottom of the 7th stripe. The length (left to right) of blue field is .76 of the overall width of the flag.

Here are the answers:

As Big As Possible: Ana wants to draw a US flag on a standard 8 1/2 by 11-inch sheet of construction paper. What are the dimensions, in inches, of the largest flag she can draw if she approximates the ratio of the width:length as 1:2?

Explanation: The sheet of paper is 8.5" x 11". If the flag is to be the right proportions, it will be twice as long as it is wide. To make it as big as possible, we should use all of one of the paper's dimensions. Since the paper is not long enough to use 8.5" for its width (that would require 8.5" x 2 = 17" of length), we should make our flag 11" long. Then it must be * 11" = 5.5" wide.
Answer: 5.5" wide x 11" long

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Red, White, and BLUE: Antonio is making a US flag with 2-inch wide stripes. If his flag has the correct ratios, what is the area of the entire rectangular blue field (including the stars)?

Explanation: If each stripe is 2" wide, then the whole flag is 13 * 2", or 26" wide. The length of the blue field is .76 * 26", or 19.76". The width of the blue field is 7 stripes or 7 * 2", which equals 14". The formula for the area of a rectangle is area = length * width.
Answer: Area of blue field = 14" * 19.76" = 276.64 square inches.

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As American As Apple "Pi": Our current flag has 50 stars, arranged in nine alternating rows of six and five stars. The original flag had 13 stars arranged in a circle, such that the outermost tip of each of the 13 stars was on the circumference of the circle. If the distance between the outermost tips of consecutive stars is 4 inches on the circumference of the circle, what is the radius of the circle, in inches? Express your answer in terms of pi.

Explanation: Since the 13 stars are equally spaced, the ratio of the distance around the circle between two stars to the circumference should be 1:13. Therefore, the circumference is 13 * 4" = 52". The radius is related to the circumference by the formula C=2π r.
Answer:
r = C/2 π
r = 52"/2 π
r = (26/π)"

If you answered the questions (even if you may have gotten something wrong), we mailed a t-shirt to you (if you provided a mailing address).

We hope you enjoyed the challenge. Check out the website often. We'll have more fun activities and challenges for you all summer long.

Thank you to Dr. Frank Serio for providing the explanations and correct answers to the math challenge questions.