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 41 km away, 24° north of west, and the second team as 33 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?
α=(90-24)+ 37=103⁰
Cosine law:
d=sqrt{ 41²+33²-2•41•33•cos103⁰}=58 km
Sine law:
33/sinβ=58/sin α
sinβ= sin α•33/58=0.554
β=sin⁻¹0.554=33.7⁰
angle=33.7⁰-24⁰=9.7⁰ (north of east)
To determine the second team's distance from the first team, we can use the Pythagorean theorem:
(a) Distance:
The first team is 41 km away from the base camp, and the second team is 33 km away from the base camp. We can use these lengths as the two legs of a right triangle.
Using the Pythagorean theorem:
Distance = √(41^2 + 33^2)
Distance ≈ 52.42 km
Therefore, the second team is approximately 52.42 km away from the first team.
(b) Direction:
To determine the direction of the second team from the first team, we can use trigonometry.
Let's consider the east direction as 0° and measure clockwise from there.
The first team is located 24° north of west, so the angle it forms with the positive x-axis is 180° - 24° = 156°.
The second team is located 37° east of north, so the angle it forms with the positive x-axis is 90° - 37° = 53°.
Since the question asks for the direction measured from due east, we need to add these two angles together:
Direction = 156° + 53°
Direction = 209°
Therefore, the second team's direction, measured from due east, is approximately 209°.
To find the answer, we can break down the given information and solve it step by step.
Let's start with the first team's location. According to the GPS tracker, the first team is 41 km away and 24° north of west. This means that if we draw a diagram, with the first team's location as the origin, the second team's location will be 41 km away in a direction that is 24° north of west.
Next, we have the second team's location. The GPS tracker shows that the second team is 33 km away and 37° east of north. If we draw a diagram with the second team's location as the origin, the first team's location will be 33 km away in a direction that is 37° east of north.
Now, to find the distance and direction when the first team uses its GPS to check the position of the second team:
(a) Distance: From the information given, we know that the first team is 41 km away from its own location and the second team is 33 km away from its own location. Since these distances are measured from their respective origins, we need to find the distance between the two origins.
Using the Pythagorean theorem, we can calculate the distance. Let's label the distance between the two origins as "d". We have a right triangle formed by the sides of 41 km, 33 km, and "d". Therefore:
d² = (41 km)² + (33 km)²
d² = 1681 km² + 1089 km²
d² = 2770 km²
d ≈ 52.63 km
So, when the first team uses its GPS to check the position of the second team, it will give a distance of approximately 52.63 km.
(b) Direction: To find the direction, we need to calculate the angle measured counterclockwise from due east to the line connecting the first team's location and the second team's location. Let's label this angle as "θ".
To solve for θ, we can use trigonometry. In this case, we have a right triangle with the opposite side as 33 km and the adjacent side as 41 km. Therefore:
tan(θ) = (33 km) / (41 km)
θ = arctan((33 km) / (41 km))
θ ≈ 39.66°
So, when the first team uses its GPS to check the position of the second team, it will give a direction of approximately 39.66°, measured counterclockwise from due east.