a ski slope at a mountain has an angle of elevation of 25.2 degrees. the vertical height of the slope is 1808 feet. how long is the ski slope. show diagram
To find the length of the ski slope, we can use trigonometry.
First, let's draw a diagram to visualize the problem. The diagram will have a right triangle, with one angle of 25.2 degrees, the given vertical height of 1808 feet, and the length of the ski slope as the hypotenuse.
/|
/ |
1808 ft / |
/ |
/ |
/ |
/θ |
/______|
Let's label the vertical height as "h" and the length of the ski slope as "l".
Now, we can use the trigonometric function "sine" to relate the angle of elevation to the ratio of the opposite side (height) to the hypotenuse, which is the ski slope length.
sin(θ) = opposite/hypotenuse
In this case, the opposite side is the vertical height (h) and the hypotenuse is the ski slope length (l).
sin(25.2 degrees) = h/l
We are given the value for h, which is 1808 feet. We need to solve for l.
To find l, we can rearrange the equation:
l = h / sin(25.2 degrees)
Now, we can substitute the given values into the equation:
l = 1808 feet / sin(25.2 degrees)
Calculating this using a scientific calculator or computer program, we find:
l ≈ 4295.8 feet
Therefore, the length of the ski slope is approximately 4295.8 feet.
To solve this problem, we can use trigonometry. Let's label the diagram as follows:
```
/|
/ |
/ |
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/ | slope
/ |
/ |
/ |
/________\
height
```
In the diagram, the vertical height is given as 1808 feet, and the angle of elevation is 25.2 degrees. We need to find the length of the ski slope.
Now, in a right-angled triangle, we can use the trigonometric function "tangent" to relate the angle of elevation to the sides of the triangle. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
In this case, the opposite side is the vertical height (1808 feet), and the adjacent side is the length of the ski slope (unknown). We can set up the equation as follows:
tan(angle) = opposite/adjacent
tan(25.2 degrees) = 1808/adjacent
To find the length of the ski slope (adjacent), we can rearrange the equation:
adjacent = opposite / tan(angle)
adjacent = 1808 / tan(25.2 degrees)
Now, let's calculate it step by step:
1. Convert the angle from degrees to radians:
angle in radians = angle in degrees * (pi/180)
25.2 degrees = 25.2 * (pi/180) radians
2. Calculate the length of the ski slope:
adjacent = 1808 / tan(25.2 * (pi/180))
Using this equation, we can substitute the values and calculate the length of the ski slope.
If you draw a right triangle to represent the slope, the slope would be the hypotenuse, and the height of the hill is the side opposite the angle 25.2 degrees (with the horizontal).
So by the definition of sin(θ), we get
sin(θ) = opposite / hypotenuse
where θ=25.2 degrees,
opposite = 1808 feet
Can you take it from here?