6.) Firefighters dug the trenches in the shape of a triangle to prevent a fire from completely destroying a forest. The lengths of the trenches were 250 feet, 312 feet, and 490 feet. Find, to the nearest degree, the smallest angle formed by the trenches. Find the area of the plot of land within the trenches, to the nearest square foot.

The smallest angle formed by the trenches is 26° and

The area of the plot of land within the trenches to the nearest foot is;

33443 ft²

Step-by-step explanation:

Here we have the firefighters made a triangular shape out of trenches surrounding the fire

The lengths of the sides of the triangle formed is given as

250 ft, 312 ft and 490 ft

We are required to find, to the nearest degree, the smallest angle formed by the trenches

We note that the smallest angle is subtended by the shortest length of the trench which is 250 ft.

Therefore,

From cosine relations, we have

a² = b² + c² - 2·b·c·cos (α)

Which gives,

250² = 312² + 490² - 2×312×490×cos (α)

cos (α) =   0.8992

cos (0.8992)⁻¹ = 25.95°   ≈ 26°

The area is given by

Where:

a, b, c are the lengths of the sides of the triangle and

s = Half the perimeter of the triangle, that us (a + b + c)/2

We put a = 250, b = 312 and c = 490

Therefore s = (250 + 312 + 490)/2 = 526

Therefore,

= 33443.025 ft² ≈ 33443 ft².

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