AB is a vertical tower 100 m from the foot of a 30-storey building CD. The angles of depression from C and E, where E is the midpoint of CD, are 35° and 14° respectively. Use mathematical tables, rounding required values to two decimal places.
Find the height of:
(a) tower AB.
(b) building CD.
Let the height of building CD be H m and the height of tower AB be h m.
BD = 100 m and E is the midpoint of CD.
Therefore, ED = \frac{\mathrm{H}}{2}.
Through A, draw AP parallel to BD. Then AP = 100 m and PD = AB = h.
In △ACP:
tan 35° = CP/AP
⇒ 0.70 = (H − h)/100
⇒ H − h = 70 … (1)
In △AEP:
tan 14° = EP/AP
⇒ 0.25 = (\frac{\mathrm{H}}{2} − h)/100
⇒ \frac{\mathrm{H}}{2} − h = 25 … (2)
Subtracting (2) from (1):
H − h − (\frac{\mathrm{H}}{2} − h) = 70 − 25
⇒ \frac{\mathrm{H}}{2} = 45
⇒ H = 90 m
Substituting in (1):
90 − h = 70
⇒ h = 20 m
Therefore, the tower is 20 m high and the building is 90 m high.
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