We can solve this problem using trigonometry. Let's break it down into steps and visualize the scenario first, then solve the problem step-by-step.

Step 1: Breakdown of the Problem

Let:

• $T$ be the top of the tower.
• $B$ be the base of the tower.
• $H_t$ be the top of the hill.
• $H_b$ be the base of the hill.
• $D$ be the horizontal distance between the tower and the hill, which is 120 ft.

The problem gives:

• The angle of elevation from $T$ to $H_t$ (top of the hill) is $27^\circ 30'$.
• The angle of depression from $T$ to $H_b$ (base of the hill) is $50^\circ 15'$.

We are asked to find the height of the hill. This includes both the height from the base to the top of the hill ($H_b$ to $H_t$) and the height of the tower.

Step 2: Drawing the Triangle

We have two right triangles here:

1. A triangle from the top of the tower ($T$) to the top of the hill ($H_t$).
2. A triangle from the top of the tower ($T$) to the base of the hill ($H_b$).

Both triangles share the same horizontal distance $D$ (120 ft).

Now, let's define:

• $h_1$ as the vertical distance from the top of the tower to the top of the hill (the height of the hill relative to the tower).
• $h_2$ as the height of the tower itself.
• The total height of the hill ($H_t$) is $h_1 + h_2$.

Step 3: Using Trigonometry

For the triangle from $T$ to $H_t$ (angle of elevation):

We use the tangent function for the angle of elevation: $\tan(27^\circ 30') = \frac{h_1}{D}$ Solving for $h_1$: $h_1 = D \cdot \tan(27^\circ 30')$

For the triangle from $T$ to $H_b$ (angle of depression):

We use the tangent function for the angle of depression: $\tan(50^\circ 15') = \frac{h_2}{D}$ Solving for $h_2$: $h_2 = D \cdot \tan(50^\circ 15')$

Step 4: Solving

Let me calculate both $h_1$ and $h_2$, and find the total height of the hill ($h_1 + h_2$).

The total height of the hill is approximately 206.75 feet. This includes both the height of the tower and the height from the top of the tower to the top of the hill.

Would you like further details or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How does changing the horizontal distance between the tower and the hill affect the total height of the hill?
2. Can you calculate the individual heights of the tower and the portion above the tower for the hill?
3. What is the significance of using the tangent function in this problem?
4. How would the solution change if the angles were given in radians instead of degrees?
5. How can we use trigonometry to find distances between other objects in real life?

Tip: Always ensure that angles in degrees are converted to radians when using trigonometric functions in most calculators or programming environments.