To find the area of triangle \( \triangle UVW \), we can use Heron's formula. Heron's formula states that if you know the lengths of all three sides of a triangle, the area \( A \) can be calculated using the formula:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( a \), \( b \), and \( c \) are the lengths of the sides, and \( s \) is the semi-perimeter given by:
\[ s = \frac{a + b + c}{2} \]
For triangle \( \triangle UVW \), we have:
- \( a = 32 \) mi
- \( b = 43 \) mi
- \( c = 17 \) mi
Step 1: Calculate the semi-perimeter \( s \).
\[ s = \frac{32 + 43 + 17}{2} = \frac{92}{2} = 46 \text{ mi} \]
Step 2: Use Heron's formula to calculate the area.
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
\[ A = \sqrt{46(46 - 32)(46 - 43)(46 - 17)} \]
Calculating each term inside the square root:
\[ 46 - 32 = 14 \]
\[ 46 - 43 = 3 \]
\[ 46 - 17 = 29 \]
Now substituting these values back into the formula:
\[ A = \sqrt{46 \times 14 \times 3 \times 29} \]
Step 3: Calculate the product:
First, find \( 46 \times 14 \):
\[ 46 \times 14 = 644 \]
Then, find \( 3 \times 29 \):
\[ 3 \times 29 = 87 \]
Now multiply both results:
\[ 644 \times 87 = 56028 \]
Step 4: Calculate the square root:
\[ A = \sqrt{56028} \approx 236.8 \text{ mi}^2 \]
Thus, the area of triangle \( \triangle UVW \) is approximately 236.8 square miles.
Since the problem asks for the answer rounded to the nearest tenth, we obtain the final answer:
\[ \boxed{236.8} \]
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