From a window 20 feet above the ground, the angle of elevation to the top of another building is 35 degrees. The distance between the buildings is 52 feet. Find the height of the building to the nearest tenth of a foot.

The lower section, [tex]\mathsf{AO}[/tex]. that's the part between the ground and the point where the building is as high as the window.
The upper section, [tex]\mathsf{BA}[/tex]. that's the part of the building above the height of the window.

To find the height of that building, simply add the length of [tex]\mathsf{AO}[/tex] (the lower section) to that of [tex]\mathsf{BA}[/tex] (the upper section).

The length of [tex]\mathsf{AO}[/tex] (the lower section) is [tex]20\; \text{feet}[/tex], same as the height of the window.
The length of [tex]\mathsf{BA}[/tex] (the upper section) can be found using the distance between the two building and that angle of elevation.

Refer to the diagram attached. [tex]\angle \mathsf{ADB}[/tex], the [tex]35^\circ[/tex] angle of elevation, is shown in green. In right triangle [tex]\triangle\mathsf{ABD}[/tex],

Segment [tex]\mathsf{AD}[/tex] is adjacent to [tex]\angle \mathsf{ADB}[/tex], while
Segment [tex]\mathsf{BA}[/tex] is opposite to [tex]\angle \mathsf{ADB}[/tex].

Besides, the length of [tex]\mathsf{AD}[/tex] is equal to [tex]52\; \text{feet}[/tex], same as the distance between the two buildings. [tex]\tan\angle \mathsf{ADB}[/tex] can help find the length of the opposite side from the length of the adjacent.

In a right triangle, the tangent of an angle is equal to the length of its opposite side divided by its adjacent side. For example, in right triangle [tex]\triangle\mathsf{ABD}[/tex],

[tex]\displaystyle \tan\angle{\mathsf{ADB}} = \frac{\text{opposite}}{\text{adjacent}} = {\frac{\mathsf{BA}}{\mathsf{AD}}}[/tex].

Therefore,

[tex]\mathsf{BA} = \mathsf{AD} \cdot \tan\angle \mathsf{ADB}[/tex].

Using a calculator,

[tex]\begin{aligned}\mathsf{BA} &= \mathsf{AD} \cdot \tan\angle \mathsf{ADB} \\ &= 52 \cdot \tan{35^\circ} \approx 36.4\; \text{feet}\end{aligned}[/tex].

Add that to the length of segment [tex]\mathsf{AO}[/tex] to find the height of this building:

[tex]\mathsf{BO} = \mathsf{BA} + \mathsf{AO} \approx 56.4\; \text{feet}[/tex].