The height of building AB is 100 feet. The height of building CD is 50 feet. The buildings are on opposite sides of an avenue that is 90 feet wide. From a point E on the avenue, the measure of the angle of elevation to B is 55º. Determine the distance from E to D to the nearest foot.
As always, draw your diagram. It should be clear that
EA = 100 cot55º
Now, (90-EA)^2 + 50^2 = ED^2
To determine the distance from point E to point D, we can use similar triangles. Let's start by drawing a diagram to visualize the situation:
```
B
/
/|
100/ |
/ |
/ |
/ |
/ |
/ |
/_______|
A 90 C
```
From the diagram, we can see that triangle ABE and triangle CDE are similar triangles. This means that the ratios of their corresponding sides are equal.
Let's assign some variables:
- Let x be the distance from point E to point D that we want to find.
- Let y be the height from point E to point B.
Since the angle of elevation from E to B is 55º, we can use trigonometry to find y:
tan(55º) = y / 90
y = 90 * tan(55º)
Now, we can set up the ratio of the corresponding sides of the similar triangles:
AB / CD = BE / DE
Plugging in the known values:
100 / 50 = (90 - y) / x
Simplifying the equation:
2 = (90 - y) / x
Substituting y with its value:
2 = (90 - 90 * tan(55º)) / x
Simplifying further:
2 = 90(1 - tan(55º)) / x
x = 90(1 - tan(55º)) / 2
Calculating the value of x using a calculator:
x ≈ 22.83
Therefore, the distance from point E to point D is approximately 22.83 feet, rounded to the nearest foot.