The summit of a mountain, 2075 m above base camp, is measured on a map to be 4585 m horizontally from the camp in a direction 32.4° west of north. Choose the x axis east, y axis north, and z axis up. What are x, y, and z?
They are distances east, north, and up.
Oh, you mean the (x,y,z) coordinates of the mountain summit?
We know z = 2075. (letting the base camp be at (0,0,0))
Now it's just a 2-d problem.
x = 4585 cos 122.4 = -4478
y = 4585 sin 122.4 = 3871
(x,y,z) = (-4478,3871,2075)
For extra points: how far is it directly from base camp to the summit?
To find the coordinates (x, y, z) of the summit, we can use trigonometry and vector components.
Let's start by setting up our coordinate system. The x-axis will be eastward, the y-axis northward, and the z-axis upward.
We are given the following information from the problem statement:
- The summit of the mountain is 2075 m above base camp, which gives us the z-coordinate.
- The summit is located 4585 m horizontally from the camp, in a direction 32.4° west of north. We can use this information to determine the x and y coordinates.
Now, let's break down the given information into vector components:
1. z-coordinate:
The summit is 2075 m above the base camp, which gives us the z-coordinate: z = 2075 m.
2. x and y coordinates:
To find the x and y coordinates, we'll use trigonometry. Since the direction is given as west of north, we'll subtract 32.4° from 90° to get the angle with respect to the positive y-axis.
Using the given angle of 32.4° and the horizontal distance of 4585 m, we can find the x and y components of this vector:
x = horizontal distance * sin(angle)
y = horizontal distance * cos(angle)
Plugging in the values, we have:
x = 4585 m * sin(32.4°)
y = 4585 m * cos(32.4°)
Calculate these values using a calculator or a computer program, and you should get:
x ≈ 2475.55 m
y ≈ 3678.47 m
So the coordinates of the summit are approximately (2475.55 m, 3678.47 m, 2075 m) in the east, north, up coordinate system.