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Answer :
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]
- 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]
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