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 find out how long Gabino and Mia should expect the entire trip to take, we need to calculate the time it takes for each leg of the journey and then sum up the times.

First, let's calculate the time it takes for Gabino and Mia to walk 2.3 km on a bearing of 305 degrees to reach point A. To do this, we use the formula:

Time = Distance / Speed

Distance = 2.3 km
Speed = 4.5 km/h

Plugging in the values, we get:

Time(A) = 2.3 km / 4.5 km/h
Time(A) ≈ 0.5111 hours

Next, let's calculate the time it takes for them to change their bearing to 64 degrees and walk 3.1 km to reach point B. Again, using the formula:

Time = Distance / Speed

Distance = 3.1 km
Speed = 4.5 km/h

Plugging in the values, we get:

Time(B) = 3.1 km / 4.5 km/h
Time(B) ≈ 0.6889 hours

Now, they need to return to their original position C. The distance from point B to point C is the same as the distance from point A to point B, which is 3.1 km.

Time(C) = Distance / Speed

Distance = 3.1 km
Speed = 4.5 km/h

Plugging in the values, we get:

Time(C) = 3.1 km / 4.5 km/h
Time(C) ≈ 0.6889 hours

Now, let's sum up the times for each leg of the journey:

Total Time = Time(A) + Time(B) + Time(C)
Total Time ≈ 0.5111 hours + 0.6889 hours + 0.6889 hours
Total Time ≈ 1.8889 hours

Therefore, Gabino and Mia should expect the entire trip to take approximately 1.8889 hours.