From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60 degrees and the angle of depression of its foot is 45 degrees. Determine the height of the tower.
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Step-by-Step Solution
Step 1: Define variables and set up the first triangle
Let AB be the building of height 7 m. Let CD be the cable tower of height H. Let BC be the distance between the building and the tower. From point A, the angle of elevation to the top of the tower (D) is 60∘, and the angle of depression to the foot of the tower (C) is 45∘. We draw a line AE parallel to BC. Thus, AE=BC and EC=AB=7 m. We need to find the total height of the tower, H=CD=CE+ED=7+h, where h=ED.
Step 2: Calculate the distance to the tower using the angle of depression
Consider the right-angled triangle △ABC. The angle of depression from A to C is 45∘. Since AE∥BC, the alternate interior angle ∠ACB is also 45∘. Using the tangent function in △ABC, we have tan45∘=BCAB. Since AB=7 m and tan45∘=1, we find that BC=7 m. Also, AE=BC=7 m.
Step 3: Calculate the height of the tower above the building
Now consider the right-angled triangle △AED. The angle of elevation from A to D is 60∘. We know AE=7 m from the previous step. Using the tangent function, tan60∘=AEED. Since tan60∘=3, we can find h=ED.
Step 4: Determine the total height of the tower
The total height of the cable tower CD is the sum of CE and ED. We know CE=7 m and we just calculated ED=h=73 m. Substitute these values to find the total height H.