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 find the distance between the peaks of the two mountains, we can use trigonometry. Specifically, we can use the tangent function.
Let's start with the angle of elevation to Mount Thomas. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the elevation of Mount Thomas above Joe's observation point (2300m - 350m), and the adjacent side is the distance to Mount Thomas from Joe's observation point.
Using the tangent function, we can write the equation as follows:
tan(22 degrees) = (2300m - 350m) / x
where x is the distance to Mount Thomas from Joe's observation point.
Now, let's move on to the angle of elevation to Mount Cook. We can use the same formula:
tan(27 degrees) = (2100m - 350m) / x
where x is the distance to Mount Cook from Joe's observation point.
Since we are looking for the distance between the peaks of the two mountains, we need to find the horizontal distance between them. To do this, we can subtract the distances of the mountains from Joe's observation point. Let's call this distance d.
So, the equation for the distance between the peaks becomes:
d = x - x
This simplifies to:
d = x
Now we have a system of two equations:
tan(22 degrees) = (2300m - 350m) / x
tan(27 degrees) = (2100m - 350m) / x
We can solve this system of equations simultaneously to find the value of x, which represents the distance between the peaks of the two mountains.