The angle of elevation of the top of a tower is 45degree from a point 10m above the water level of a lake.From the same point, the angle of depression of its image in the lake is 60degree.Find the height of the t
Draw a diagram. Let point M be where the image of the top of the tower appears.
P = point of observation
Q = point on water below P
T = top of tower
B = base of tower at water level
so, the height h of the tower is BT
So, now we have
PM = 10/√3
Since light reflects at equal angle, angle TMB is also 60°
Now we have
h/MB = tan60° = √3
But, MB = QB-QM - h-10-10/√3
so,
h/(h-10-10/√3) = √3
h = 10(2+√3) ≈ 37.32 m
To find the height of the tower, we can use the trigonometric ratios of angles in a right triangle.
Let's assume the height of the tower is represented by 'h'.
Given that the angle of elevation from the point to the top of the tower is 45 degrees, we can construct a right triangle as follows:
/
/|
/ |
height(h) / |
/ |
/_______|
|---------|
distance
Now, let's consider the angle of depression of the image of the tower in the lake. From the same point, the angle of depression is 60 degrees.
We can construct another triangle using this information:
______
| |
| |
height(h) | |
| |
|--------| distance
river
We can see that the height of the tower is the same as the depth of the image in the lake.
Based on this, we can form the following equation:
tan(45°) = tan(60°) + h/distance
Since the angle of elevation and depression are complementary angles (45° + 60° = 105°), we can substitute the values into the equation:
1 = tan(105°) + h/distance
Now, we need to find the value of tan(105°). We know that tan(90° + 15°) = -cot(15°) = -tan(75°).
So, we can substitute -tan(75°) into the equation:
1 = -tan(75°) + h/distance
The distance between the point and the tower, as well as the distance between the point and the image, is 10m. So, we can substitute this value into the equation:
1 = -tan(75°) + h/10
To find the value of h, we can rearrange the equation:
h/10 = 1 + tan(75°)
h/10 = 1 + tan(45° + 30°)
h/10 = 1 + (tan(45°) + tan(30°))/(1 - tan(45°)tan(30°))
h/10 = 1 + (1 + √3)/(1 - √3)
h/10 = (2 - √3)/(1 - √3)
h = 10 * (2 - √3)/(1 - √3)
Simplifying the expression, we get:
h = 10 * (2 - √3)/(1 - √3)
h ≈ 10 * (2 - 1.732)/(1 - 1.732)
h ≈ 10 * (0.268)/(0.268)
h ≈ 10 meters
Therefore, the height of the tower is approximately 10 meters.
To find the height of the tower, let's break down the problem into smaller components.
Let's assume that the height of the tower is h meters.
From the given information, we have two triangles: the triangle formed by the top of the tower, the point above the water level, and the point where the observer is standing, and the triangle formed by the top of the tower, its image in the lake, and the same point where the observer is standing.
First, let's consider the triangle formed by the top of the tower, the point above the water level, and the observer's point. Since the angle of elevation from the observer's point to the top of the tower is 45 degrees, we can use tan 45 degrees to find the ratio of the height of the tower to the distance from the observer's point to the tower.
tan(45°) = h / x (x is the distance from the observer's point to the tower)
Since tan(45°) = 1, the equation simplifies to h = x.
Now, let's consider the triangle formed by the top of the tower, its image in the lake, and the observer's point. Since the angle of depression from the observer's point to the image of the tower is 60 degrees, we can use tan 60 degrees to find the ratio of the height of the tower to the distance from the observer's point to the tower.
tan(60°) = h / (10 + x) (10 is the height of the observer's point above the water level; (10 + x) represents the total height from the water level to the top of the tower)
Simplifying the equation, we have √3 = h / (10 + x).
Now, we can solve these two equations simultaneously to find the height of the tower.
h = x
√3 = h / (10 + x)
By substituting the value of h from the first equation into the second equation, we get:
√3 = x / (10 + x)
Now, we can solve the equation for x:
√3(10 + x) = x
Expanding the equation, we get:
10√3 + √3x = x
Isolating the x term, we have:
√3x - x = -10√3
(x - √3x) = -10√3
Factor out x: x(1 - √3) = -10√3
Divide both sides by (1 - √3): x = (-10√3) / (1 - √3)
Multiply the numerator and denominator by (1 + √3): x = (-10√3)(1 + √3) / (1 - √3)(1 + √3)
Simplifying further, we get: x = (-10√3 - 30) / -2
x = 15(√3 + 1)
Now, substituting this value of x into the equation h = x, we can find the height of the tower:
h = 15(√3 + 1)
Therefore, the height of the tower is approximately 25.98 meters.