Answer:
The distance between the buildings is:
D = 413.3 ft
Step-by-step explanation:
Ok, here we have two triangle rectangles, such that these triangle rectangles share a common vertex which is Dale.
One of the triangles, let's call it triangle A, has the catheti:
Distance between Dale and Building A = d1
Height of Building A = 464 ft
The other triangle (triangle B) has the catheti:
Distance between Dale and Building B = d2
Height of building B = 321ft
We want to find the distance between the buildings, which is equal to d1 + d2.
Then we want to find d1 and d2.
For triangle A:
We know that the angle at the vertex where Dale is standing, is 75°
Then the adjacent cathetus is d1, and the opposite cathetus is the height of building A.
Remember that:
Tan(θ) = (opposite cathetus)/(adjacent cathetus)
Then:
Tan(75°) = (464 ft)/(d1)
d1 = (464 ft)/(Tan(75°)) = 124.3 ft
Now for triangle B, the angle at the vertex where Dale is standing is 48°.
from this vertex, we have d2 as the adjacent cathetus and the height of build B as the opposite cathetus.
Then:
Tan(48°) = 321ft/d2
d2 = (321ft)/(Tan(48°)) = 289ft
Then the distance between the buildings is:
D = d1 + d2 = 124.3 ft + 289ft = 413.3 ft
