Points A and B are on the same horizontal line withthe foot of a hill and the angles of depression of thesepoints from the top of the hill are 30.2° and 22.5°,respectively. If the distance between A and B is 75.0m, what is the height of the hill?

To solve this problem, we will use the concept of trigonometry, specifically the tangent of an angle in a right triangle.

Step 1: Identify the triangles We have two right triangles here. Triangle TBA and Triangle TAC. T is the top of the hill, B and A are points on the ground, and C is the point on the ground directly below T.

Step 2: Use the tangent of the angles The tangent of an angle in a right triangle is equal to the opposite side divided by the adjacent side. We can write this as:

tan(θ) = opposite/adjacent

Step 3: Apply to our triangles For Triangle TBA, the angle is 30.2° and the opposite side is the height of the hill (h), and the adjacent side is the distance BA. So we have:

tan(30.2°) = h/BA

For Triangle TAC, the angle is 22.5° and the opposite side is also the height of the hill (h), and the adjacent side is the distance AC. So we have:

tan(22.5°) = h/AC

Step 4: Use the fact that AC = AB + BC The distance AC is equal to the distance AB plus the distance BC. We know that AB = 75m, so we can substitute this into our second equation to get:

tan(22.5°) = h/(75m + BC)

Step 5: Solve the equations Now we have two equations with two unknowns (h and BC), which we can solve simultaneously.

From the first equation, we can express h as h = BA * tan(30.2°)

Substitute h into the second equation, we get BA * tan(30.2°) = (75m + BC) * tan(22.5°)

Solving for BC, we get BC = (BA * tan(30.2°) / tan(22.5°)) - 75m

Substitute BC back into the equation for h, we get the height of the hill h = BA * tan(30.2°) - BC * tan(22.5°)

Step 6: Calculate the height Now we can plug in the given values to calculate the height of the hill.