From a mountain 1780 ft. high, the angle of depression of a point on the nearer shore of a river is 48deg40mins and of point directly across on the opposite side is 22deg20mins. What is the width of the river between the two points?
The height of the mountain is represented by the ver. side of a rt triangle. The line of sight of the 48
deg. angle represents the hypotenuse.
The hor. side is X.
A 2nd rt triangle
is formed by the line of sight of the
22 deg. angle, and its' ver. side is also the height of the mountain. the
hor. side = X + W where W is the width of the river.
Tan(48,40min) = 1780/X, X = 1565.6 Ft
Tan(22,20min) = 1780/(1565.6+W).
1565.6 + W = 1780/Tan(22,20min),
W = 2767.3 Ft. = Width of river.
from the top of a mountain 532 m higher than a nearly river, the angle of depression of a point P on the closer bank of the river is 52.6o and the angle of depression of a point Q directly opposite. on P in the other side is 34.5o points P,Q and the foot of the mountain are on the same horizontal line. find the distance across the river from P to Q
To solve this problem, we can use the trigonometric concept of angles of depression. Let's consider the given information:
Height of the mountain (H) = 1780 ft
Angle of depression of the nearer shore point (A) = 48°40'
Angle of depression of the point directly across on the opposite side (B) = 22°20'
We need to find the width of the river (W) between the two points.
Step 1: Find the tangent of angle A
tan(A) = opposite/adjacent
tan(48°40') = H/W
Step 2: Solve for W
W = H / tan(A)
W = 1780 ft / tan(48°40')
Step 3: Convert angle A to decimal degrees
48°40' = 48 + (40/60) = 48.6667°
Step 4: Calculate the tangent of angle A using its decimal value
W = 1780 ft / tan(48.6667°)
Step 5: Calculate the width of the river (W)
W ≈ 1432.26 ft
Therefore, the width of the river between the two points is approximately 1432.26 feet.
To find the width of the river, we can use trigonometry.
Let's first draw a diagram to represent the situation described:
```
|\
| \
| \ d
| \
| \
|____\
x
```
In this diagram, the mountain is represented as a vertical line of height 1780 ft. The line labeled "d" represents the width of the river, and the line labeled "x" represents the horizontal distance between the two points on the opposite sides of the river.
From the given information, we know that the angle of depression from the mountain top to the nearer point on the shore is 48°40', and the angle of depression to the point directly across is 22°20'.
Let's calculate the distance "x" first:
Using trigonometry, we can use the tangent function:
tan(angle) = opposite/adjacent
For the angle of depression 48°40', the opposite side is the height of the mountain (1780 ft), and the adjacent side is "x". So we have:
tan(48°40') = 1780 ft / x
Similarly, for the angle of depression 22°20', the opposite side is the height of the mountain (1780 ft), and the adjacent side is "x + d" (because the width of the river is added to the horizontal distance). So we have:
tan(22°20') = 1780 ft / (x + d)
Now we have a system of two equations with two unknowns (x and d):
1. tan(48°40') = 1780 ft / x
2. tan(22°20') = 1780 ft / (x + d)
We can solve this system of equations to find the values of "x" and "d".
1. Rewrite the first equation:
x = 1780 ft / tan(48°40')
2. Substitute the value of "x" in the second equation:
tan(22°20') = 1780 ft / ((1780 ft / tan(48°40')) + d)
Now we can solve this equation for "d".
Note: I will calculate the result for you.