Orienteering is a sport that involves navigating your way to a series of checkpoints with the aid of a map and compass. The goal is to find each point on the map as quickly as possible. A typical compass used for orienteering is divided into 360 degree where 0 degrees is north and a bearing is measured clockwise from north.

A.) Gabino and Mia are taking part in an orienteering course. They are instructed to walk 2.3 km on a bearing of 305 degrees to reach point A. There they are supposed to change their bearing to 64 degrees and walk 3.1 km to reach point B. Next they are supposed o return to their original position C. If they walk at an average pace of 4.5 km/h, how long should they expect the entire trip to take?

To calculate the time it will take for Gabino and Mia to complete the entire trip, we need to determine the total distance they will walk and divide it by their average pace.

First, let's calculate the distance they will walk from their starting point C to point A. They are instructed to walk 2.3 km on a bearing of 305 degrees. To find the horizontal distance (east-west) they will cover, we use the cosine function: cos(305) * 2.3 km. Similarly, to find the vertical distance (north-south) they will cover, we use the sine function: sin(305) * 2.3 km.

Horizontal distance from C to A = cos(305) * 2.3 km
Vertical distance from C to A = sin(305) * 2.3 km

Next, let's calculate the distance they will walk from point A to point B. They are instructed to walk 3.1 km on a bearing of 64 degrees. Again, we use the cosine function to find the horizontal distance and the sine function to find the vertical distance.

Horizontal distance from A to B = cos(64) * 3.1 km
Vertical distance from A to B = sin(64) * 3.1 km

Now, let's calculate the distance they will walk from point B to their starting point C. To find this distance, we need to calculate the horizontal and vertical distances in the opposite direction from point A to point C.

Horizontal distance from B to C = -cos(64) * 3.1 km
Vertical distance from B to C = -sin(64) * 3.1 km

To find the total distance traveled, we sum up the absolute values of all the distances:

Total distance = |C to A| + |A to B| + |B to C|
= |cos(305) * 2.3 km| + |sin(305) * 2.3 km| + |cos(64) * 3.1 km| + |sin(64) * 3.1 km| + |-cos(64) * 3.1 km| + |-sin(64) * 3.1 km|

Finally, we divide the total distance by their average pace of 4.5 km/h to get the expected time for the entire trip:

Time = Total distance / Average pace
= (|cos(305) * 2.3 km| + |sin(305) * 2.3 km| + |cos(64) * 3.1 km| + |sin(64) * 3.1 km| + |-cos(64) * 3.1 km| + |-sin(64) * 3.1 km|) / 4.5 hours

By using appropriate calculator software or applying trigonometric functions on the angles, you can evaluate this expression to find the expected time for the entire trip.