From a window 33.0 ft above the​ street, the angle of elevation to the top of the building across the street is 45.0​° and the angle of depression to the base of this building is 17.0​°. Find the height of the building across the street.

To find the height of the building across the street, we can break the problem into two parts:

1. **From the window to the top of the building** (using the angle of elevation).
2. **From the window to the base of the building** (using the angle of depression).

We'll use trigonometry to solve this. Let:

- \( h \) = total height of the building across the street.
- \( d \) = horizontal distance between the two buildings.
- The window height is 33.0 feet above the ground.

### 1. Using the angle of elevation to the top of the building:

The angle of elevation to the top of the building is 45°. For this part, the triangle formed involves the height from the window to the top of the building and the horizontal distance \( d \).

- The height from the window to the top of the building = \( h_{top} = h - 33 \).
- From the angle of elevation, we have:
  \[
  \tan(45^\circ) = \frac{h_{top}}{d}
  \]
  Since \( \tan(45^\circ) = 1 \), we get:
  \[
  1 = \frac{h - 33}{d} \quad \Rightarrow \quad d = h - 33.
  \]

### 2. Using the angle of depression to the base of the building:

The angle of depression to the base of the building is 17°. For this part, the triangle formed involves the height of the window above the ground (33 ft) and the same horizontal distance \( d \).

- From the angle of depression, we have:
  \[
  \tan(17^\circ) = \frac{33}{d}.
  \]
  Solving for \( d \):
  \[
  d = \frac{33}{\tan(17^\circ)}.
  \]

### Now, equate the two expressions for \( d \):

\[
h - 33 = \frac{33}{\tan(17^\circ)}.
\]
Solve for \( h \):

\[
h = \frac{33}{\tan(17^\circ)} + 33.
\]

Let me calculate this for you.The total height of the building across the street is approximately 140.94 feet.

Would you like more details, or do you have any further questions?

Here are five related questions to consider:
1. How would the height change if the window were located higher above the street?
2. What would happen if the angle of elevation increased to 60°?
3. How can you use the Pythagorean theorem to verify the horizontal distance?
4. How would the solution change if the angle of depression were measured incorrectly?
5. What if the angle of elevation and depression were the same?

**Tip:** Always double-check the units (e.g., degrees vs. radians) when using trigonometric functions.

From a window 33.0 ft above the street, the angle of elevation to the top of the building across the street is 45.0° and the angle of depression to the base of this building is 17.0°. Find the height of the building across the street.

Learn how to calculate the height of a building across the street using trigonometric concepts. This problem involves angles of elevation (45°) and depression (17°) with a step-by-step solution suitable for Grades 10-12.

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