The angle of elevation from the top of one building to the top of another is [tex]$26^{\circ}$[/tex]. If the shorter building is 400 feet tall and the buildings are 1500 feet apart, how tall is the other building?

We are given:

- The height of the shorter building is [tex]$400$[/tex] feet.
- The horizontal distance between the buildings is [tex]$1500$[/tex] feet.
- The angle of elevation from the top of the shorter building to the top of the taller building is [tex]$26^\circ$[/tex].

Step 1. Calculate the difference in height.

The tangent of an angle in a right triangle is given by

[tex]$$
\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}.
$$[/tex]

In this situation, the "opposite side" is the difference in height between the two buildings, and the "adjacent side" is the horizontal distance between them. Therefore,

[tex]$$
\tan 26^\circ = \frac{\text{height difference}}{1500}.
$$[/tex]

To find the height difference, multiply both sides by [tex]$1500$[/tex]:

[tex]$$
\text{height difference} = 1500 \tan 26^\circ.
$$[/tex]

Using the value

[tex]$$
\tan 26^\circ \approx 0.48773258856586144,
$$[/tex]

we have

[tex]$$
\text{height difference} \approx 1500 \times 0.48773258856586144 \approx 731.5988828487922 \text{ feet}.
$$[/tex]

Step 2. Find the height of the taller building.

The height of the taller building is the sum of the height of the shorter building and the height difference:

[tex]$$
\text{taller building} = 400 + 731.5988828487922 \approx 1131.598882848792 \text{ feet}.
$$[/tex]

Final Answer:

The other building is approximately

[tex]$$
\boxed{1131.6 \text{ feet tall}}.
$$[/tex]