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The angles of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at a distance 9 ft and 16 ft respectively are complementary angles. Then the height of the tower is (SSC CGL 2nd Sit. 2012)
- a)9 ft
- b)12 ft
- c)16 ft
- d)144 ft
Correct answer is option 'B'. Can you explain this answer?
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The angles of elevation of the top of a tower standing on a horizontal...
In ΔABC
tan α = h/9 ... (1)
In ΔABD
tan β = h/16
α + β = 90° (given)
β = 90 – α
Since, tan β = h/16
From eqn. (1) and (2)
tan α = h/9 ... (1)
In ΔABD
tan β = h/16
α + β = 90° (given)
β = 90 – α
Since, tan β = h/16
From eqn. (1) and (2)
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The angles of elevation of the top of a tower standing on a horizontal...
Given:
- The angles of elevation of the top of a tower from two points on a line passing through the foot of the tower are complementary angles.
- The distance of the first point from the foot of the tower is 9 ft.
- The distance of the second point from the foot of the tower is 16 ft.
To find:
The height of the tower.
Let's assume the height of the tower is 'h' ft.
From the given information, we can form a right triangle with the tower as the vertical side, the horizontal plane as the base, and the line passing through the foot of the tower as the hypotenuse.
Let's consider the first point. The angle of elevation from this point is the angle between the horizontal plane and the line joining the first point to the top of the tower. Let's denote this angle as θ.
From the right triangle, we can write the following trigonometric equation:
tan(θ) = h/9
Similarly, considering the second point, the angle of elevation from this point is the angle between the horizontal plane and the line joining the second point to the top of the tower. Let's denote this angle as α.
From the right triangle, we can write the following trigonometric equation:
tan(α) = h/16
Given that the angles of elevation are complementary, we have:
θ + α = 90°
Now, we can solve these equations to find the value of 'h'.
Solving the equations:
tan(θ) = h/9
tan(α) = h/16
From the given condition, we have:
tan(θ) = cot(α)
Using the trigonometric identity:
cot(α) = 1/tan(α)
Substituting the values, we get:
h/9 = 1/(h/16)
h/9 = 16/h
Cross multiplying, we have:
h^2 = 9 * 16
h^2 = 144
Taking the square root of both sides, we get:
h = √144
h = 12 ft
Therefore, the height of the tower is 12 ft (option B).
- The angles of elevation of the top of a tower from two points on a line passing through the foot of the tower are complementary angles.
- The distance of the first point from the foot of the tower is 9 ft.
- The distance of the second point from the foot of the tower is 16 ft.
To find:
The height of the tower.
Let's assume the height of the tower is 'h' ft.
From the given information, we can form a right triangle with the tower as the vertical side, the horizontal plane as the base, and the line passing through the foot of the tower as the hypotenuse.
Let's consider the first point. The angle of elevation from this point is the angle between the horizontal plane and the line joining the first point to the top of the tower. Let's denote this angle as θ.
From the right triangle, we can write the following trigonometric equation:
tan(θ) = h/9
Similarly, considering the second point, the angle of elevation from this point is the angle between the horizontal plane and the line joining the second point to the top of the tower. Let's denote this angle as α.
From the right triangle, we can write the following trigonometric equation:
tan(α) = h/16
Given that the angles of elevation are complementary, we have:
θ + α = 90°
Now, we can solve these equations to find the value of 'h'.
Solving the equations:
tan(θ) = h/9
tan(α) = h/16
From the given condition, we have:
tan(θ) = cot(α)
Using the trigonometric identity:
cot(α) = 1/tan(α)
Substituting the values, we get:
h/9 = 1/(h/16)
h/9 = 16/h
Cross multiplying, we have:
h^2 = 9 * 16
h^2 = 144
Taking the square root of both sides, we get:
h = √144
h = 12 ft
Therefore, the height of the tower is 12 ft (option B).
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The angles of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at a distance 9 ft and 16 ft respectively are complementary angles. Then the height of the tower is (SSC CGL 2nd Sit. 2012)a)9 ftb)12 ftc)16 ftd)144 ftCorrect answer is option 'B'. Can you explain this answer?
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