The Angle of depression of Boat A from the top of a cliff which is 32 m high is 24 degree 15 minutes .The angle of depression of Boat B from the same point is 18 degree 12 minutes .Find distance between the two side with diagram
h = 32 m.
A = 24.25o
B = 18.2o
Tan24.25 = h/X1 = 32/X1
X1*Tan24.25 = 32
X1 = 32/Tan24.25 = 71 m.=Hor. distance
between boat A and the cliff.
Tan18.2 = h/(X1+X2) = 32/(71+X2)
(71+X2)*Tan18.2 = 32
71+X2 = 32/Tan18.2 = 97.33
X2 = 97.33-71 = 26.3 m=Distance between
boat A and B.
Thank you for this idea 😍
Thanks it helps a lot
Ah, a question about boats and angles! Let's dive right in, shall we?
First, let's draw a lovely diagram to help us visualize the situation. Imagine we have a cliff, which is 32 meters high, and we're standing at the top. We have Boat A somewhere below, and Boat B somewhere else, both on the same side.
```
Cliff (32 m)
_________________________
|/////////////
| //////////
| ///////
| /////
|_________//______
O O
Boat A Boat B
```
Now that we have our beautiful diagram, let's start solving the problem!
From the top of the cliff, the angle of depression to Boat A is 24 degrees 15 minutes. That means if we draw a line of sight straight down to Boat A, the angle between this line and the horizontal level is 24 degrees 15 minutes.
Similarly, the angle of depression to Boat B is 18 degrees 12 minutes. So the angle between the line of sight straight down to Boat B and the horizontal level is 18 degrees 12 minutes.
Now, to find the distance between the two boats, we need to use the concept of trigonometry. Specifically, we can use tangent, since we have the angle of depression and the height of the cliff as reference points.
Let's call the distance between the cliff and Boat A "x" and the distance between the cliff and Boat B "y".
For Boat A:
We have the angle of depression of 24 degrees 15 minutes, which is equivalent to approximately 0.424 radians.
Using tangent, we can say that:
tan(0.424) = 32 / x
Solving for x, we get:
x = 32 / tan(0.424)
Similarly, for Boat B:
We have the angle of depression of 18 degrees 12 minutes, which is approximately 0.317 radians.
Using tangent again, we can say that:
tan(0.317) = 32 / y
Solving for y, we get:
y = 32 / tan(0.317)
And there you have it! The distance between Boat A and Boat B is approximately x - y.
Remember, my friend, to always have a sense of humor when tackling math problems. Happy calculating!
To find the distance between the two boats, we can use trigonometry. Let's represent the distance between the two boats as "d."
Let's begin by drawing a diagram to visualize the problem.
1. Draw a horizontal line to represent the top of the cliff.
2. Label a point "P" on top of the cliff.
3. Draw two vertical lines, one representing Boat A and the other Boat B, below the point P.
4. Label the distance from the top of the cliff to Boat A as "h" (which is 32 meters), and the angle of depression of Boat A as 24 degrees 15 minutes.
5. Label the distance from the top of the cliff to Boat B as "h" (which is also 32 meters), and the angle of depression of Boat B as 18 degrees 12 minutes.
6. Label the distance between Boat A and Boat B as "d."
Now that we have the diagram, let's calculate the value of "d" using trigonometry.
For Boat A:
Using the angle of depression and the height of the cliff, we can determine the distance from the base of the cliff to Boat A by using the concept of tangent.
Using the trigonometric identity: tan(angle) = opposite/adjacent, we have:
tangent(24 degrees 15 minutes) = Boat A's distance (opposite) / height of the cliff (adjacent).
Convert the angle of depression from degrees + minutes to decimal form:
24 degrees + 15 minutes = 24 + 15/60 = 24.25 degrees.
Now we can calculate the distance to Boat A:
tangent(24.25 degrees) = Boat A's distance (opposite) / 32 meters (adjacent).
Let's denote the distance to Boat A as "x":
tan(24.25 degrees) = x / 32.
We can solve for x by multiplying both sides by 32:
x = 32 * tan(24.25 degrees).
Using a calculator, we find that x ≈ 14.88 meters.
For Boat B:
Using a similar approach, let's calculate the distance to Boat B.
First, convert the angle of depression for Boat B from degrees + minutes to decimal form:
18 degrees + 12 minutes = 18 + 12/60 = 18.20 degrees.
Now we can calculate the distance to Boat B:
tan(18.20 degrees) = Boat B's distance (opposite) / 32 meters (adjacent).
Let's denote the distance to Boat B as "y":
tan(18.20 degrees) = y / 32.
Solving for y by multiplying both sides by 32:
y = 32 * tan(18.20 degrees).
Using a calculator, we find that y ≈ 11.42 meters.
Finally, to find the distance between the two boats, subtract the distances:
d = x - y = 14.88 meters - 11.42 meters.
Therefore, the distance between the two boats is approximately 3.46 meters.