Two geological field teams are working in a remote area. A global positioning system (GPS) tracker at their base camp shows the location of the first team as 42 km away, 21° north of west, and the second team as 29 km away, 37° east of north. When the first team uses its GPS to check the position of the second team, what does it give for the second team's (a) distance from them and (b) direction, measured from due east?

D = 29km[153o] - 42km[143o]

a. X = 29*cos153 - 42*cos143 = 7.70 km.
Y = 29*sin153 - 42*sin143 = -12.1 km.

D^=X^2 + Y^2 = 7.7^2 + (-12.1)^2=205.94
D = 14.35 km. = The dist. between the 2
teams.

b. tan A = Y/X = -12.1/7.70 -1.57142857
A = -57.53o = 57.53o South of East.

To solve this problem, we can break it down into two parts. First, we need to find the coordinates of both teams using their respective distances and directions. Then, we can calculate the distance and direction between the two teams.

Let's start with the first team. According to the GPS tracker, the first team is 42 km away and 21° north of west. We can convert this information into Cartesian coordinates.

To find the x-coordinate (west-east direction), we can use the formula:
x = distance * cos(direction)

Substituting in the values:
x1 = 42 km * cos(21°)

Calculating the value:
x1 ≈ 38.405 km

To find the y-coordinate (north-south direction), we can use the formula:
y = distance * sin(direction)

Substituting in the values:
y1 = 42 km * sin(21°)

Calculating the value:
y1 ≈ 15.283 km

So, the first team's coordinates are approximately (38.405 km, 15.283 km).

Now let's move on to the second team. The GPS tracker shows that the second team is 29 km away and 37° east of north. Again, we can use the same formulas to find the Cartesian coordinates.

Calculating the x-coordinate:
x2 = 29 km * sin(37°)

Calculating the value:
x2 ≈ 17.613 km

Calculating the y-coordinate:
y2 = 29 km * cos(37°)

Calculating the value:
y2 ≈ 21.537 km

So, the second team's coordinates are approximately (17.613 km, 21.537 km).

Now, we can find the distance and direction between the two teams.

To calculate the distance, we can use the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Substituting in the values:
distance = sqrt((17.613 km - 38.405 km)^2 + (21.537 km - 15.283 km)^2)

Calculating the value:
distance ≈ 23.969 km

So, the distance between the two teams is approximately 23.969 km.

To calculate the direction, we can use the direction formula:
direction = atan((y2 - y1) / (x2 - x1))

Substituting in the values:
direction = atan((21.537 km - 15.283 km) / (17.613 km - 38.405 km))

Calculating the value:
direction ≈ -25.42°

Since the direction is measured from due east, we need to convert it to a positive angle by adding 90°:
direction ≈ 64.58°

So, the direction of the second team from the first team, measured from due east, is approximately 64.58°.