From the top of height y m the angle of depression of a Channel swimmer at X is 30°. She swims directly towards the base of the cliff such that 1 minute later, at Y, the angle of depression from the top of the cliff to the swimmer is 50°. Given that her speed between X and Y is 0.75m/s, find the height of the cliff.
.75m/s * 60s = 45m
Draw a diagram. It will show that
y cot 30° - y cot 50° = 45
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To find the height of the cliff, we can use the concept of trigonometry and the given information about the angles of depression and the swimmer's speed. Let's break down the problem step by step:
Step 1: Draw a diagram
Draw a rough diagram of the scenario, labeling the known measurements and angles. Label the top of the cliff as point T, the swimmer's position at time X as point X, the swimmer's position at time Y as point Y, and the base of the cliff as point B. Also, label the height of the cliff as h and the distance from X to Y as d.
Step 2: Identify the right-angled triangles
In the diagram, you should have two right-angled triangles. Triangle TXY has a right angle at point X, and triangle TBY has a right angle at point B.
Step 3: Define the trigonometric relationships
In triangle TXY, we can define the following relationships:
- tan(30°) = h / d (since the angle of depression from T to X is 30°)
- tan(50°) = h / (d + 60) (since the angle of depression from T to Y is 50° and the swimmer moves 60 meters in 1 minute)
Step 4: Solve the equations
First, rearrange the equation for the first relationship:
h = d * tan(30°)
Now, rearrange the equation for the second relationship:
h = (d + 60) * tan(50°)
Step 5: Equate the two equations and solve for h
Set the two expressions for h equal to each other:
d * tan(30°) = (d + 60) * tan(50°)
Now, we can solve this equation for d:
d * tan(30°) = d * tan(50°) + 60 * tan(50°)
d * tan(30°) - d * tan(50°) = 60 * tan(50°)
d * (tan(30°) - tan(50°)) = 60 * tan(50°)
d = (60 * tan(50°)) / (tan(30°) - tan(50°))
Step 6: Calculate the value of d
Using a calculator, substitute the values of the tangents:
d ≈ (60 * 1.19) / (0.58 - 1.19)
d ≈ (71.4) / (-0.61)
d ≈ -117.05
Note: The negative sign indicates that d is measured in the opposite direction from the swimmer's movement towards the base of the cliff. Therefore, we can ignore the negative sign and consider d as a positive value of approximately 117.05 meters.
Step 7: Calculate the height of the cliff
Now, substitute the value of d into the equation for the height of the cliff:
h = d * tan(30°)
h ≈ 117.05 * 0.58
h ≈ 67.82 meters
Therefore, the height of the cliff is approximately 67.82 meters.
To find the height of the cliff, we can use trigonometry and set up a system of equations.
Let's denote the height of the cliff as h meters and the distance between X and Y as d meters.
From the given information:
1. The angle of depression from the top of the cliff to the swimmer at X is 30°.
2. The angle of depression from the top of the cliff to the swimmer at Y, 1 minute later, is 50°.
3. The swimmer's speed between X and Y is 0.75 m/s.
First, we can calculate the distance between X and Y using the swimmer's speed:
d = speed × time = 0.75 m/s × 60 s = 45 m
Next, we can find the horizontal distance between the swimmer at X and the base of the cliff (BX). Since the swimmer is swimming directly towards the base of the cliff, BX = d.
Now, let's consider the right-angled triangle formed by the swimmer at X, the top of the cliff, and the base of the cliff. Using trigonometry (tangent function), we can write the following equation:
tan(30°) = h / d
Simplifying this equation, we have:
h = d × tan(30°)
We can plug in the value of d:
h = 45 m × tan(30°)
Now we have the height of the cliff, h. Let's calculate this value:
h ≈ 25.98 m
Therefore, the height of the cliff is approximately 25.98 meters.