Answer:
Answer in terms of a trigonometric function :
[tex]h = \frac{20tan(19.9)}{0.4-tan(19.9)}[/tex]
Answer in figures
[tex]h = 190.5 feet[/tex]
Step-by-step explanation:
Consider the sketch attached below to better understand the problem.
Let x be the distance between point B and the base of the water tower.
[tex]tan (19.9)=\frac{h}{50+x} ---------------equation 1\\[/tex]
[tex]tan (21.8)=\frac{h}{x}----------------equation 2[/tex]
From equation 2,
[tex]x =\frac{h}{tan21.8}=\frac{h}{0.4}[/tex]
substituting the value of x into equation 1, we get
[tex]tan (19.9)=\frac{h}{50+(\frac{h}{0.4} )}[/tex]
[tex]tan 19.9=h\times \frac{0.4}{20+h}[/tex]
cross multiplying,
[tex]20\times tan(19.9) +htan(19.9)=0.4h[/tex]
[tex]20tan(19.9)= h(0.4-tan(19.9))[/tex]
[tex]h= \frac{20tan (19.9)}{0.4-tan (19.9)}[/tex]
The height of the tower is [tex]h= \frac{20tan (19.9)}{0.4-tan (19.9)}[/tex] in terms of the trig function "Tan"
The equation can simply be evaluated to get the answer in figures since the angles are given in the question
The height of the tower in feet is 190.63 ft
The situation forms two right angle triangle.
Right triangle has one of its angle as 90 degrees and the sides can be found using trigonometric ratios. The height of the tower is the opposite side of the triangle formed.
The height of the tower can be found as follows;
let
h = height of tower
y = distance of the tower to the closer point (point B)
Therefore,
tan ∅ = opposite / adjacent
tan 21.8 = h / y
h = y tan 21.8
tan 19.9 = h / y + 50
y = (y + 50) tan 19.9
Therefore,
y tan 21.8 = (y + 50) tan 19.9
y tan 21.8 = y tan 19.9 + 50 tan 19.9
y tan 21.8 - y tan 19.9 = 50 tan 19.9
y (tan 21.8 - y tan 19.9) = 18.099747324
y(0.39997146407 - 0.36199494648) = 18.099747324
y = 18.099747324/0.03797651807
y = 476.603655202
y = 476.60 ft
Therefore,
h = y tan 21.8
h = 476.60 × 0.39997146407
h = 190.626399778
h = 190.63 ft
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