July 3, 2017

To celebrate Independence Day, students attending a summer camp painted the American flag mural, show here, with the dimensions in centimeters.  If the area of each of the stars is 22.2 cm2, how many square centimeters of the flag is painted blue?

 

To determine the area of the flag painted blue we can find the area of the entire blue region then subtract the combined area of the stars. The area of the blue portion of the flag containing the stars is 74 × 104.5 = 7733 cm2.  There are 50 stars on the flag, each with an area of 22.2 cm2.  The total combined area of the stars is 22.2 × 50 = 1110 cm2.  That means the area of the flag that is painted blue is 7733 – 1110 = 6623 cm2.

 

Assuming each of the stripes has the same thickness, what is the positive difference, in square meters, between the area of the flag covered in white stars and the area of the flag covered in white stripes? Express your answer as a decimal to the nearest tenth.

In the previous problem we determined the area of the flag covered in white stars to be 1110 cm2. Now we need to find the area of the flag covered in white stripes. First we notice that not all the white stripes are the same length.  Let’s look at the stripes in two groups. The first group of stripes spans the entire width of the flag. There are a total of 6 stripes in this section (3 red and 3 white). That means that half of the total area of this section is covered with white stripes. The height of this section of the flag is 137.4 – 74 = 63.4 cm. So the area of this section is 63.4 × 261.3 = 16,566.42 cm2. Half of that area, 8283.21 cm2, is covered in white stripes. The second group of stripes doesn’t span the entire width of the flag, just 261.3 – 104.5 = 156.8 cm. The height of this section of stripes  is 74 cm. So the total area of the flag covered by this group of 7 stripes is 74 × 156.8 = 11,603.2 cm2. But only 3 of the 7 stripes are painted white. That means only (3/7)(11,603.2) = 4972.8 cm2 of this portion of the flag is covered in white stripes. The total area of the flag covered in white stripes is 8283.21 + 4972.8 = 13,256.01 cm2. Therefore, the positive difference between the area of the flag covered in white stripes and the area of the flag covered in white stars is 13,256.01 – 1110 = 12,146.01 ¸ (100)2 = 12,146.01 ¸ 10,000 = 1.2 m2. (Calculations may be slightly different if you calculate the thickness of each stripe as 137.4 ¸ 13 = 10.56923, and calculate the area using this figure. )

 

Again, assuming each of the stripes has the same thickness, what percent of the flag mural is not painted white?

The entire flag has an area of 137.4 × 261.3 = 35,902.62 cm2.  Only the stars and some of the stripes are pointed white. From the previous problems, the total area of the flag that is painted white is 1110 + 13,256.01 = 14,366.01 cm2.  That means that 35,902.62 − 14,366.01 = 21,536.61 cm2 is the area of the flag not painted white.  This is equivalent to 21,536.61/35,902.62 = 0.5998617928 ≈ 60% of the flag mural.

 

Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2
contains ONLY PROBLEMS.