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A canoe is paddled away from a dock directly toward a buoy that is 5 km away. A...

Dec 20, 2024

Solution by Steps

step 1

We can use the Law of Cosines to find the distance from the dock to the canoe's final position. The formula is given by:

c^2 = a^2 + b^2 - 2ab \cos(C)

, where

a = 5

km (distance to the buoy),

b = 3

km (distance from the buoy to the canoe's final position), and

C = 12^\circ

step 2

Plugging in the values into the formula:

c^2 = 5^2 + 3^2 - 2 \cdot 5 \cdot 3 \cdot \cos(12^\circ)

step 3

Calculating the squares:

c^2 = 25 + 9 - 30 \cdot \cos(12^\circ)

step 4

Now, we calculate

\cos(12^\circ)

and substitute it back:

c^2 = 34 - 30 \cdot 0.9781

(approximately)

step 5

Continuing the calculation:

c^2 = 34 - 29.343 \approx 4.657

step 6

Finally, taking the square root gives us

c \approx \sqrt{4.657} \approx 2.16

km. Rounding to the nearest tenth, we get

2.2

Answer

2.2 km

Key Concept

Law of Cosines

Explanation

The Law of Cosines allows us to find the length of a side in a triangle when we know the lengths of the other two sides and the included angle. In this case, it helped us determine the distance from the dock to the canoe's final position.