Answer:

In the first scenario, let's solve for the distance between the two ships, given the distance from observation pole A to the airplane (400 ft), the distance from observation pole B to the airplane (300 m), and the angle at observation pole A (45°).

Using the Law of Cosines:

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

Where:

c = distance between the two ships (unknown)

a = distance from observation pole A to the airplane (400 ft)

b = distance from observation pole B to the airplane (300 m)

C = angle at observation pole A (45°)

Converting 300 m to feet (1 m ≈ 3.281 ft):

b = 300 m * 3.281 ft/m = 984.3 ft

Plugging in the values into the equation:

c^2 = (400 ft)^2 + (984.3 ft)^2 - 2 * (400 ft) * (984.3 ft) * cos(45°)

Simplifying:

c^2 = 160,000 ft^2 + 968,176.49 ft^2 - 787,424 ft^2

c^2 = 340,752.49 ft^2

Taking the square root of both sides:

c ≈ √(340,752.49 ft^2) ≈ 584.2 ft

Therefore, the distance between the two ships is approximately 584.2 feet.

Now, moving on to the second scenario, where we need to solve the triangle with side lengths a = 25, b = 10, and c = 20 using the Law of Cosines.

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

Plugging in the values:

(20)^2 = (25)^2 + (10)^2 - 2 * (25) * (10) * cos(C)

400 = 625 + 100 - 500 * cos(C)

400 = 725 - 500 * cos(C)

500 * cos(C) = 725 - 400

500 * cos(C) = 325

cos(C) = 325 / 500

cos(C) ≈ 0.65

C ≈ cos^(-1)(0.65)

C ≈ 48.19°

Therefore, the angle C is approximately 48.19°.