To solve such problems we must know about Tangent (Tanθ).
What is Tangent (Tanθ) ?
The tangent or tanθ in a right angle triangle is the ratio of its perpendicular to its base. it is given as,
[tex]\rm{Tangent(\theta) = \dfrac{Perpendicular}{Base}[/tex]
where,
θ is the angle,
Perpendicular is the side of the triangle opposite to the angle θ,
The base is the adjacent smaller side of the angle θ.
The height of the mountain is 1.495 miles.
Given to us
- AB = 15 miles
- ∠CAD = 3.5°
- ∠CBD = 9°
In ΔCAD,
[tex]\rm{ tan(\angle A) = \dfrac{Perpendicular}{Base}[/tex]
[tex]tan(\angle A) = \dfrac{CD}{AD}[/tex]
[tex]tan(3.5^o) = \dfrac{CD}{15+x}[/tex]
[tex]CD=tan(3.5^o) \times(x+15)[/tex]
In ΔCBD,
[tex]\rm{ tan(\angle B) = \dfrac{Perpendicular}{Base}[/tex]
[tex]tan(\angle B) = \dfrac{CD}{BD}[/tex]
[tex]tan(9^o) = \dfrac{CD}{x}[/tex]
[tex]CD=tan(9^o) \times(x)[/tex]
Equating CD,
[tex]tan(3.5^o) \times (x+15) = tan(9^o) \times x\\\\(x+15)= \dfrac{tan(9^o) }{tan(3.5^o) }\times x\\\\x+15=2.59\ x\\15 = 2.59x-x\\15= 1.59x\\x = 9.44[/tex]
Substituting the value of x
[tex]CD=tan(9^o) \times(x)[/tex]
[tex]CD=tan(9^o) \times(9.44)\\CD= 1.495[/tex]
Hence, the height of the mountain is 1.495 miles.
Learn more about Tangent (Tanθ) function:
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