The distance from an observer on the plain to the top of a nearby mountain is 5.3 km, and the angle between this line and the horizonatal is 8.4 degrees. How tall is the mountain?
sin 8.4o = height/5.3
solve for height (in km).
0.77
To find the height of the mountain, we can use trigonometry.
First, let's draw a diagram to better understand the situation:
```
Mountain
/|
/ |
/ | h (height of the mountain)
Observer / |
o-------- / |
distance / |
(5.3 km) / |
/ |
/ θ (angle between
Horizontal / observer's line of sight
/ and the horizontal)
/
/
/
/
/
```
In this case, we have a triangle where the distance from the observer to the mountain is the base, the height of the mountain is the opposite side, and the angle theta (θ) is given.
Now, we can use the trigonometric relationship of tangent (tan) in a right triangle to calculate the height of the mountain.
The tangent of an angle (θ) in a right triangle is defined as the ratio of the length of the opposite side (height of the mountain) to the length of the adjacent side (distance from observer to the mountain).
Mathematically, we can write:
tan(θ) = (height of the mountain) / (distance from observer to the mountain)
Rearranging the equation to solve for the height of the mountain (h), we have:
(height of the mountain) = (tan(θ)) * (distance from observer to the mountain)
Now let's substitute the given values into the equation:
(height of the mountain) = (tan(8.4 degrees)) * (5.3 km)
To solve this, we need to convert the angle from degrees to radians, as most trigonometric functions in programming languages require angles to be in radians.
Radians = (degrees * π) / 180
Converting the angle:
θ (in radians) = (8.4 degrees * π) / 180
Now we have the angle θ in radians, which we can use in the formula to find the height of the mountain:
(height of the mountain) = (tan(θ (in radians))) * (5.3 km)
Using a calculator or a programming language with trigonometric functions, calculate the tangent of θ (in radians), multiply it by the distance from the observer to the mountain (5.3 km), and you will get the height of the mountain.