Illustrate,label and solve the problem.the angle of depression of boat A from the top of a cliff which is 24"15'.the angle of depression of boat B from the same point is 18"12".find the distance between the two boats.
So draw the diagram and review your basic trig functions.
You will notice that it is impossible to solve, since you don't give the height of the cliff. But, if the angles of boats A and B are a and b, and the height is h, then you can find the distance x between the boats using
x = h cot(b) - h cot(a)
To solve this problem, we can use trigonometric principles such as the tangent function to find the distances between the boats.
1. Let's label the given information:
- Angle of depression of boat A from the top of the cliff: 24° 15'
- Angle of depression of boat B from the same point: 18° 12'
2. Now, let's draw a diagram to visualize the problem. Draw a cliff and label its height h. Then draw two boats, A and B, on the same horizontal line as the cliff, as shown:
A B
_________
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__|_________|________
Cliff h
3. We need to find the distance between the two boats. Let's label the distance between the observer on the cliff and boat A as x and the distance between the observer on the cliff and boat B as y.
4. Using the tangent function, we can set up the following equations:
- For boat A:
tan(24° 15') = h / x
- For boat B:
tan(18° 12') = h / y
5. Now we have a system of equations:
- Equation 1: tan(24° 15') = h / x
- Equation 2: tan(18° 12') = h / y
6. Convert the given angles into decimal degrees:
- 24° 15' = 24 + (15/60) = 24.25°
- 18° 12' = 18 + (12/60) = 18.2°
7. Substitute the decimal degrees into the equations:
- Equation 1: tan(24.25°) = h / x
- Equation 2: tan(18.2°) = h / y
8. To solve for h, we can set up an equation using the Law of Sines, which states:
- h / sin(90°) = x / sin(65.75°)
9. Simplify the equation:
- h = x * sin(65.75°)
10. Substitute the value of h into Equation 2 and solve for y:
- tan(18.2°) = (x * sin(65.75°)) / y
11. Rearrange the equation and solve for y:
- y = (x * sin(65.75°)) / tan(18.2°)
12. Now we have two equations to solve for the distances x and y:
- Equation 1: tan(24.25°) = (x * sin(65.75°)) / y
- Equation 2: y = (x * sin(65.75°)) / tan(18.2°)
13. Substitute Equation 2 into Equation 1:
- tan(24.25°) = (x * sin(65.75°)) / ((x * sin(65.75°)) / tan(18.2°))
14. Simplify the equation:
- tan(24.25°) = tan(18.2°) * x
15. Solve for x:
- x = tan(24.25°) / tan(18.2°)
16. Use a calculator to find the decimal value of x by dividing the tangent of 24.25° by the tangent of 18.2°.
17. After finding the decimal value of x, you can calculate the distances x and y by substituting the x value into either Equation 1 or Equation 2.
18. Finally, calculate the distance between the two boats by subtracting the values of x and y: Distance = y - x.
Now you have the step-by-step explanation and procedure to solve the problem. Perform the calculations to get the final answer.