To solve for the height of the flagpole using similar triangles, we can set up a proportion based on the corresponding heights and shadow lengths.
First, convert Myk's height from feet and inches to just inches:
- Myk's height = 5 feet 2 inches = \(5 \times 12 + 2 = 60 + 2 = 62\) inches.
- Myk's shadow = 84 inches.
- Flagpole's shadow = 31 feet = \(31 \times 12 = 372\) inches.
Now we can set up the proportion based on the heights and shadows:
\[ \frac{\text{Height of Myk}}{\text{Shadow of Myk}} = \frac{\text{Height of Flagpole}}{\text{Shadow of Flagpole}} \]
Substituting the known values:
\[ \frac{62}{84} = \frac{h}{372} \]
Cross-multiplying gives:
\[ 62 \times 372 = 84 \times h \]
Calculating \(62 \times 372\):
\[ 62 \times 372 = 23064 \]
Now, we can solve for \(h\):
\[ 84h = 23064 \]
\[ h = \frac{23064}{84} \]
Calculating \(h\):
\[ h \approx 274.14 \text{ inches} \]
To convert inches to feet, divide by 12:
\[ h \approx \frac{274.14}{12} \approx 22.85 \text{ feet} \]
Rounding to the nearest foot, the height of the flagpole is:
23 feet.
Therefore, the answer is 23 feet.