a tower that is a 200 meters is leaning to one side. from a certain point on that side, the angle of elevation to the top of the tower is 70 degree. From a point 55 meters closer to the tower, the angle of elevation is 85 degree. Determine the acute angle from the horizontal at which the tower is leaning?
First observation point A
second observation point B
top of tower P
bottom of tower Q
In triangle ABP,
angle A = 70 --- given
angle PBA = 95 --- exterior angle to 85°
angle APB = 15°
AB = 55m --- given
by the sine law
BP/sin70 = 55/sin15
BP = 55sin70/sin15
in triangle BPQ
PQ=200
angle PBQ = 85
BP = 55sin70/sin15
sinQ/BP= sin85/200
sinQ = BPsin85/200
= (55sin70/sin15)(sin85)/200 = .9946..
angle Q = 84.0658°
or appr 84°
Well, it seems like the tower is really embracing its rebellious side and leaning to one side! Let's help it find its balance, shall we?
First, let's break down the given information. We have two angles of elevation: one of 70 degrees and another of 85 degrees. The difference between the two points is 55 meters, creating a nice triangle for us to work with.
Now, to determine the acute angle from the horizontal at which the tower is leaning, we need to find the angle at the top of the triangle.
Using our trigonometry superpowers, we can draw a right-angled triangle with the hypotenuse representing the height of the tower (200 meters) and the adjacent side representing the horizontal distance from the certain point (let's call this distance 'x').
Using the tangent function, we can write the first equation:
tan(70) = (200 / x)
Next, let's zoom in a little closer to the tower, where the distance from the tower is now 'x - 55'. We can now create another right-angled triangle, similar to the previous one, but with a different angle of elevation (85 degrees).
Using the tangent function again, we write the second equation:
tan(85) = (200 / (x - 55))
Now, we have a system of equations to solve:
1. tan(70) = (200 / x)
2. tan(85) = (200 / (x - 55))
Once you figure out the solution to this mathematical puzzle, you'll find the acute angle from the horizontal at which the tower is leaning. Happy solving!
To determine the acute angle from the horizontal at which the tower is leaning, we can use trigonometry.
Let's denote the distance from the certain point to the tower as 'x'. From the information given, we have the following triangle:
A
/|
/ |
/ |x
/ |
/ |
/ |
/θ |
/_____|
B C
Where:
- Point A represents the top of the tower
- Point B represents the certain point on the side where the angle of elevation is 70 degrees
- Point C represents the certain point on the side where the angle of elevation is 85 degrees
- AB = 200 meters (height of the tower)
- BC = x meters
- AC = x - 55 meters (since the distance from point C to the tower is 55 meters less than the distance from point B)
Using trigonometry, we can establish the following relationships:
In triangle ABC:
tanθ = AB / BC
In triangle ACB:
tan(85°) = AB / (BC + 55)
Now, let's solve for 'x' by setting up and solving these two equations:
tan(70°) = 200 / x
tan(85°) = 200 / (x + 55)
Cross multiplying, we get:
x = 200 / tan(70°)
x + 55 = 200 / tan(85°)
Evaluating these expressions, we find:
x ≈ 250.322 meters
x + 55 ≈ 368.135 meters
Now that we have the values for 'x' and 'x + 55', we can calculate the acute angle from the horizontal using the tangent function:
tan(angle) = (x - (x + 55)) / 200
tan(angle) = -55 / 200
Simplifying further, we find:
angle ≈ arctan(-55/200) ≈ -15.3 degrees
Therefore, the acute angle from the horizontal at which the tower is leaning is approximately 15.3 degrees in the opposite direction.
To determine the acute angle at which the tower is leaning, we can use the concept of trigonometry and the given information.
Let's denote the height of the tower as "h" and the distance from the certain point on the side to the tower as "x".
From the information given, we have two right triangles:
In the first right triangle, with the angle of elevation of 70 degrees, the opposite side is "h" and the adjacent side is "x". We can use the tangent function because the tangent of an angle in a right triangle is defined by the ratio of the opposite side to the adjacent side:
tan(70 degrees) = h / x
In the second right triangle, with the angle of elevation of 85 degrees, the opposite side is "h" and the adjacent side is "x + 55" (since the point is 55 meters closer to the tower). Again, we can use the tangent function:
tan(85 degrees) = h / (x + 55)
Now we have two equations with two unknowns. We can solve these equations simultaneously to find the values of "h" and "x".
Dividing the first equation by the second equation:
tan(70 degrees) / tan(85 degrees) = (h / x) / (h / (x + 55))
tan(70 degrees) / tan(85 degrees) = (x + 55) / x
Now we can substitute the values of the tangents of 70 degrees and 85 degrees:
0.8391 / 11.4301 = (x + 55) / x
0.0733 = (x + 55) / x
Cross multiplying:
0.0733x = x + 55
0.0733x - x = 55
-0.9267x = 55
x = 55 / -0.9267
x ≈ -59.38 meters
Since the distance cannot be negative, it means that the tower is leaning away from the certain point on the side. Therefore, we take the absolute value of "x":
x ≈ 59.38 meters
Now, to determine the acute angle from the horizontal at which the tower is leaning, we can use the tangent function again:
tan(theta) = h / x
tan(theta) = h / 59.38
To find the value of theta, we can take the inverse tangent (arctan) of both sides:
theta = arctan(h / 59.38)
However, we need to find the value of "h" to calculate theta. To do so, we can substitute the value of "x" in one of the initial equations (let's use the first one):
tan(70 degrees) = h / 59.38
Now, solve for "h":
h ≈ 59.38 * tan(70 degrees)
h ≈ 186.45 meters
Substitute this value of "h" into our expression for theta:
theta = arctan(186.45 / 59.38)
theta ≈ 73.75 degrees
Therefore, the acute angle from the horizontal at which the tower is leaning is approximately 73.75 degrees.