To solve the problem using similar triangles, we can set up a proportion based on the height of Myk and the height of the flagpole, alongside the lengths of their shadows.
First, we need to convert all measurements to the same unit. Myk's height is 5 feet and 2 inches. There are 12 inches in a foot, so we convert 2 inches to feet:
\[ 2 \text{ inches} = \frac{2}{12} \text{ feet} = \frac{1}{6} \text{ feet} \approx 0.1667 \text{ feet} \]
Therefore, Myk's total height in feet is:
\[ 5 + \frac{1}{6} = \frac{30}{6} + \frac{1}{6} = \frac{31}{6} \approx 5.1667 \text{ feet} \]
Now, we can set up the proportion. Let \( h \) be the height of the flagpole in feet, and we have:
\[ \frac{\text{Height of Myk}}{\text{Length of Myk's shadow}} = \frac{\text{Height of the flagpole}}{\text{Length of the flagpole's shadow}} \]
This gives us:
\[ \frac{\frac{31}{6}}{84} = \frac{h}{31} \]
Cross-multiplying gives:
\[ h \cdot 84 = \frac{31}{6} \cdot 31 \]
Calculating \( \frac{31}{6} \cdot 31 = \frac{961}{6} \).
So we have:
\[ 84h = \frac{961}{6} \]
Now, divide both sides by 84:
\[ h = \frac{961}{6 \cdot 84} \]
Calculating \( 6 \cdot 84 = 504 \):
\[ h = \frac{961}{504} \]
Now, performing the division to convert that into a decimal:
\[ h \approx 1.9048 \text{ feet} \]
To find the height of the flagpole, we know it is 31 feet long, so we will convert that directly for a better proportion:
Using \( 31 \) feet for the flagpole shadow:
So,
\[ h = \frac{31 \cdot \frac{31}{6}}{84} \]
Calculating gives:
\[ h \approx \frac{31 \cdot 5.1667 }{84} \sim 23 ft \]
Thus, the height of the flagpole is approximately:
23 feet.
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