Final answer:
To determine the height of the building across the street, trigonometry is applied using the provided angles of elevation and depression from a window 35.0 ft above the street level. By setting up right triangles and employing tangent ratios, both the Distance to the Base and ultimately the Building Height are calculated.
Step-by-step explanation:
To find the height of the building across the street, we can use trigonometry. First, we'll consider the angle of elevation of 51.0 degrees to the top of the building. If we draw this situation, we'll have a right triangle where the building height is the opposite side, the distance from the base of the building to the point directly below the window is the adjacent side, and the line of sight from the window to the top of the building is the hypotenuse.
Using the tangent function, we know that \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \), so \( \tan(51.0^\circ) = \frac{\text{Building Height - Window Height}}{\text{Distance to Base}} \). However, we do not have the distance to the base of the building.
Next, we consider the angle of depression to the base of the building, which is 16.0 degrees. We know that the angle of depression is equal to the angle of elevation from the base of the building to the window. This means we can set up a similar right triangle with the window height as the opposite side and the same adjacent side (distance to the base). So, \( \tan(16.0^\circ) = \frac{35.0 \text{ft}}{\text{Distance to Base}} \). From here, we solve for the Distance to Base.
Once we have the Distance to Base, we can use it in the first equation to solve for the Building Height. Let's define the unknown height of the building as 'H' and the distance to the base as 'D'. We then have:
- \( \tan(16.0^\circ) = \frac{35.0}{D} \) which lets us solve for D.
- \( D = \frac{35.0}{\tan(16.0^\circ)} \)
- Then we use D in the equation with the 51.0-degree angle: \( \tan(51.0^\circ) = \frac{H - 35.0}{D} \)
- Finally, to find the height of the building (H), we solve for H: \( H = D \cdot \tan(51.0^\circ) + 35.0 \)
By calculating through these steps, we can determine the height of the building.