Pre Calc HELP high school/college

Question

1. To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle 75° from the horizontal. An observer D = 580 m away measures the angle of elevation to the spot of light to be 45°. Find the height h of the cloud cover. (Round your answer to the nearest meter.)
2. To find the distance across a small lake, a surveyor has taken the measurements shown. Find the distance across the lake using this information. (Assume points A and B are exactly along the shoreline, and that a = 2.92 miles and b = 3.11 miles. Round your answer to two decimal places.)

Answer 1

Using the fact that every triangle has 180°, θ = 180 - 45 - 75 = 60°

Using the Law of Sines:

meaning a = 580sin(75°)/sin(60°) = 646.906

Then, using the right triangle created by the altitude "h", we can say:

sin(45°) = h/646.906 so

h = 646.906sin(45°)

h = 457.43 meters

For the 2nd problem, use the Law of Cosines to find the distance between points A and B (side "c"):

c = √(2.92² + 3.11² - 2(2.92)(3.11)cos(40.3) = √4.3466 = 2.08485 mi.

Then use the Law of Sines to find angle A: sin(A)/2.92 = sin(40.3)/2.08485 so sin(A) = 0.90588 making angle A = sin^-1(0.90588) = 64.94°.

Then use the right triangle created from the perpendicular distance from point C to line AB to get:

sin(64.94°) = d/3.11 making the distance across the lake (d) = 2.82 mi