Question 1179789: 300 ft from the base of a mountain the top and the base of a tower on its summit is sighted @ 27degree 43’ & 24degree 25’ respectively. At the base of the mountain the angles of elevation are 39degree & 35degree respectively. Find the height of the tower?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step using trigonometry.
**1. Diagram:**
* Draw a horizontal line representing the ground.
* Mark a point A on the ground, 300 ft from the base of the mountain.
* Draw a vertical line representing the mountain. Let the base of the mountain be B and the top be C.
* Draw a tower on top of the mountain. Let the top of the tower be D.
* From point A, draw lines to B, C, and D.
* Draw a vertical line from the base of the mountain (B) to the top of the mountain (C). Also draw a vertical line from the base of the mountain to the top of the tower (D).
**2. Given Information:**
* Distance from A to the base of the mountain (AB) = 300 ft
* Angle of elevation from A to the base of the tower (∠BAB) = 24°25'
* Angle of elevation from A to the top of the tower (∠BAD) = 27°43'
* Angle of elevation from B to the base of the tower (∠EBC) = 35°
* Angle of elevation from B to the top of the tower (∠EBD) = 39° where E is a point on the horizontal line drawn from A to the base of the mountain.
**3. Convert Angles to Decimal Degrees:**
* 24°25' = 24 + 25/60 ≈ 24.4167°
* 27°43' = 27 + 43/60 ≈ 27.7167°
* 35°
* 39°
**4. Calculate the Height of the Mountain (BC):**
* In triangle ABC, tan(24.4167°) = BC / AB
* BC = AB * tan(24.4167°)
* BC = 300 * tan(24.4167°)
* BC ≈ 300 * 0.4539
* BC ≈ 136.17 ft
**5. Calculate the Height of the Mountain Plus Tower (BD):**
* In triangle ABD, tan(27.7167°) = BD / AB
* BD = AB * tan(27.7167°)
* BD = 300 * tan(27.7167°)
* BD ≈ 300 * 0.5244
* BD ≈ 157.32 ft
**6. Calculate the Height of the Tower (CD):**
* CD = BD - BC
* CD = 157.32 - 136.17
* CD ≈ 21.15 ft
**7. Calculate the height of the mountain using the angles of elevation at B:**
* Let the height of the mountain be h.
* tan(35°) = h/ distance from B to a point on the horizontal line from A.
* Let the distance from B to a point on the horizontal line be x.
* tan(39°) = (h+tower)/x.
* tan(35°) = h/x
* x = h/tan(35°)
* tan(39°) = (h+tower)/(h/tan(35°))
* tan(39°) = (h+tower)*tan(35°)/h
* h*tan(39°) = h*tan(35°) + tower*tan(35°)
* h(tan(39°)-tan(35°)) = tower*tan(35°)
* tower = h(tan(39°)-tan(35°))/tan(35°)
* tower = h(0.8098-0.7002)/0.7002
* tower = h(0.1096)/0.7002
* tower = 0.1565h.
**8. Calculate the height of the mountain using the first set of angles:**
* BC = 136.17 ft.
**9. Calculate the height of the tower using the first set of angles:**
* CD = 21.15 ft.
**10. Substitute BC into the equation for the tower:**
* tower = 0.1565*136.17
* tower = 21.3193
**Answer:**
The height of the tower is approximately 21.15 feet.
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