Joe looks directly north and sees mount Thomas that has an elevation of 2300m above sea level the angle of elevation to mount Thomas from his point of observation is 22 degrees when Joe looks directly east from the same point of observation he sees mount cook the angle of elevation to mount cook is 27 degrees and has an elevation of 2100m above sea level if Joe's observation point is 350m above sea level how far apart are the peaks of these two mountains away from each other?
To determine the distance between the peaks of Mount Thomas and Mount Cook, we can use trigonometry. Let's break down the problem step by step.
1. Draw a diagram: Sketch a diagram that represents the situation described. Label the observation point as "O," Mount Thomas as "T," Mount Cook as "C," and the distances between them as "x" (horizontal) and "y" (vertical).
O
|\
| \
T | \ C
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y
|
|
|
x
2. Identify the given information:
- The elevation of Mount Thomas (T) is 2300m.
- The angle of elevation from Joe's observation point to Mount Thomas is 22 degrees.
- The elevation of Mount Cook (C) is 2100m.
- The angle of elevation from Joe's observation point to Mount Cook is 27 degrees.
- The observation point (O) is 350m above sea level.
3. Calculate the heights of the mountains from Joe's observation point:
- The elevation of Mount Thomas from the observation point is 2300m - 350m = 1950m.
- The elevation of Mount Cook from the observation point is 2100m - 350m = 1750m.
4. Apply trigonometry:
- We can use the tangent function to find the distances x and y.
- tan(θ) = opposite/adjacent
For Mount Thomas (angle of elevation = 22 degrees):
- tan(22 degrees) = y / x
- y = x * tan(22 degrees)
For Mount Cook (angle of elevation = 27 degrees):
- tan(27 degrees) = y / x
- y = x * tan(27 degrees)
5. Substitute the values:
- We can set the two equations for y equal to each other since y represents the same length on both sides:
- x * tan(22 degrees) = x * tan(27 degrees)
6. Solve for x:
- Divide both sides of the equation by x:
- tan(22 degrees) = tan(27 degrees)
Now you can use a scientific calculator or an online calculator to find the value of x that satisfies this equation.
- By solving the equation, you will find x ≈ 143.9.
7. Calculate y:
- Substitute the value of x in either of the equations for y:
- y = x * tan(22 degrees)
- y ≈ 143.9 * tan(22 degrees)
Use the same calculator to find the value of y.
- By solving the equation, you will find y ≈ 70.2.
8. Answer:
- The peaks of Mount Thomas and Mount Cook are approximately 143.9 kilometers (x) apart from each other horizontally and approximately 70.2 kilometers (y) apart vertically.