A tower 150 m high is situated at the top of a hill. At a point 650 m down the hill, the angle
between the surface of the hill and the line of sight to the top of the tower is 12 deg 30 min.
Find the inclination of the hill to a horizontal plane.
do the choices happen to be:
A. 5°54'
B. 7°10'
C. 6°12'
D. 7°50
?
From your:
cos (12.5° + θ) / sin(12.5°) = 650/150
why not just:
cos(12.5+θ) = 650sin12.5/150 = .9379...
12.5+θ = 20.2973...
θ = 7.797
or their answer of 7° 50'
Let
A = top of the tower,
B = base of the tower and
C = point of observation
=> ∠ ACB = 12.5°
Let θ = angle of inclination of the hill to the horizontal plane.
=> ∠ CAB = 90° - (12.5° + θ)
Applying sine rule to the triangle ABC,
sin [90° - (12.5° + θ)] / 650 = sinθ / 150
=> cos (12.5° + θ) / sinθ = 650/150
=> cos(12.5°) cotθ - sin(12.5°) = 13/3
=> (0.9763) cotθ = 4.3333 + 0.2164
=> cotθ = 4.6602
=> θ = 12.11°
- Kesha
wait, what's the answer choices???
1/3
because if so, it would be D)
Reasoning:
Applying sine rule to the triangle ABC,
sin [90° - (12.5° + θ)] / 650 = sin(12.5°) / 150
=> cos (12.5° + θ) / sin(12.5°) = 650/150
=> cot(12.5°) cosθ - sinθ = 13/3
=> 4.5107 cosθ - sinθ = 4.3333
=> 4.5107 (1 - tan^2 (θ/2)) - 2tan(θ/2) = 4.3333 (1 + tan^2 (θ/2))
=> 8.841 tan^2 (θ/2) + 2tan(θ/2) - 0.1774 = 0
=> tan(θ/2) = 1/(17.682) [- 2 + √(4 + 6.2736)]
=> tan(θ/2) = 0.06816
=> θ/2 = 3.9°
=> θ = 7.8°.
Answer: D).
Well, that's a lot of numbers and angles! It's like a math riddle, but I'll do my best to help you out.
First, let's break it down. We have a tower 150 m high, and we're standing 650 m down the hill. The angle between the surface of the hill and the line of sight to the top of the tower is 12 degrees 30 minutes.
To find the inclination of the hill to a horizontal plane, we need to find the angle between the hill and the horizontal plane. So let's think about it - if we draw a triangle, with the tower as the top vertex, the hill as the base, and the line of sight as the hypotenuse, we can see that the angle we're looking for is the complement of the angle of 12 degrees 30 minutes.
Now, let's do the math. To convert 12 degrees 30 minutes into decimal degrees, we can divide the 30 minutes by 60 and add it to the 12 degrees. So we have (12 + 30/60) degrees = 12.5 degrees.
The complement of an angle is 90 degrees minus that angle. So the complementary angle to 12.5 degrees is 90 - 12.5 = 77.5 degrees.
Therefore, the inclination of the hill to a horizontal plane is 77.5 degrees. Phew, math can be a real hill-a-rious challenge sometimes!
To find the inclination of the hill to a horizontal plane, we can use trigonometry.
Let's denote the height of the tower as h = 150 m and the distance from the point down the hill to the tower as d = 650 m. We also have the angle between the surface of the hill and the line of sight to the top of the tower as 12° 30'.
First, we need to find the distance from the point down the hill to the bottom of the tower. We can do this by using trigonometry:
sin(12° 30') = h / distance_to_bottom_of_tower
Rearranging the equation, we get:
distance_to_bottom_of_tower = h / sin(12° 30')
Now, we can find the total distance from the bottom of the tower to the top of the hill by adding the distance to the bottom of the tower with the distance from the bottom of the tower to the hill:
total_distance = distance_to_bottom_of_tower + d
Finally, we can find the inclination of the hill to a horizontal plane:
inclination = atan(h / total_distance)
Plug in the given values and compute the inclination using a calculator or software, and you will get the answer.