1. A balloon starts at position (1.3, -1.1, 3.1), where the units are in kilometers. If it is blown by the wind 16 kilometers along the two angles theta = 175 degrees and phi = -11 degrees, what is the distance from the balloon to the origin? How high is the balloon off the ground?
in x y plane the balloon moves 16 cos 11 km = 15.7 km
so x displacement = 15.7 cos 175 = - 15.6
so final x= 1.3 -15.6 = 14.3
y displacement = 15.7 sin 175 = 1.37
so final y = -1.1 + 1.37 = 0.27
z displacement = -16 sin 11 = -3.05
so final z = 3.1 - 3.05 = 0.05 (about 50 meters above crash)
r = sqrt (14.3^2 + 0.27^2 + 0.05^2)
To find the distance from the balloon to the origin, we can use the concept of spherical coordinates. Spherical coordinates represent points in three-dimensional space using two angles, theta and phi, and a radial distance. The two angles, theta and phi, represent the direction to the point from the origin, and the radial distance represents the distance from the origin to the point.
Given the starting position of the balloon (1.3, -1.1, 3.1) in kilometers, we can convert this Cartesian coordinate to spherical coordinates.
To convert the Cartesian coordinates (x, y, z) to spherical coordinates (r, theta, phi), we can use the following formulas:
r = sqrt(x^2 + y^2 + z^2)
theta = arccos(z / r)
phi = arctan(y / x)
Let's calculate the spherical coordinates for the starting position (1.3, -1.1, 3.1):
r = sqrt((1.3)^2 + (-1.1)^2 + (3.1)^2)
= sqrt(1.69 + 1.21 + 9.61)
= sqrt(12.51)
≈ 3.54 km
theta = arccos(3.1 / 3.54)
= arccos(0.8757)
≈ 29.03 degrees
phi = arctan((-1.1) / 1.3)
= arctan(-0.846)
≈ -40.36 degrees
Now, we can use the given wind displacement of 16 kilometers along the two angles theta = 175 degrees and phi = -11 degrees to find the new position of the balloon.
To calculate the new position, we first need to convert the wind displacement from spherical coordinates to Cartesian coordinates.
x = r * sin(theta) * cos(phi)
y = r * sin(theta) * sin(phi)
z = r * cos(theta)
Let's calculate the new Cartesian coordinates:
x = 16 * sin(175 degrees) * cos(-11 degrees)
≈ -8.68 km
y = 16 * sin(175 degrees) * sin(-11 degrees)
≈ -5.20 km
z = 16 * cos(175 degrees)
≈ 14.88 km
Now, we can calculate the distance from the balloon to the origin using the new Cartesian coordinates (x, y, z).
distance = sqrt((0 - x)^2 + (0 - y)^2 + (0 - z)^2)
= sqrt((-8.68)^2 + (-5.20)^2 + (14.88)^2)
≈ sqrt(75.47 + 27.04 + 221.45)
≈ sqrt(323.96)
≈ 17.99 km
Therefore, the distance from the balloon to the origin is approximately 17.99 kilometers.
To determine how high the balloon is off the ground, we need to calculate the vertical distance between the starting position and the new position of the balloon.
height = z - (-1.1)
≈ 14.88 - (-1.1)
≈ 15.98 km
Therefore, the balloon is approximately 15.98 kilometers off the ground.