To find the height of the flagpole using the principle of similar triangles, we can set up a proportion based on the height and the length of the shadows of Myk and the flagpole.
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Convert Myk's height into inches: Myk's height = 5 feet 2 inches = (5 x 12) + 2 = 60 + 2 = 62 inches.
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Myk's shadow = 84 inches.
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Flagpole's shadow = 31 feet = 31 x 12 = 372 inches.
Now, we set up the proportion:
\[ \frac{\text{Height of Myk}}{\text{Shadow of Myk}} = \frac{\text{Height of Flagpole}}{\text{Shadow of Flagpole}} \] \[ \frac{62}{84} = \frac{h}{372} \]
where \(h\) is the height of the flagpole. Now we can cross-multiply to solve for \(h\):
\[ 62 \times 372 = 84 \times h \]
Calculating \(62 \times 372\):
\[ 62 \times 372 = 23064 \]
Then:
\[ 84h = 23064 \]
Now, divide both sides by 84 to find \(h\):
\[ h = \frac{23064}{84} \approx 274.8571 \text{ inches} \]
Now, convert the height from inches to feet:
\[ h = \frac{274.8571}{12} \approx 22.9048 \text{ feet} \]
Rounding to the nearest foot, the height of the flagpole is approximately 23 feet.
So the answer is 23 feet.