An explorer is caught in a whiteout (in which the snowfall is so thick that the ground cannot be distinguished from the sky) while returning to base camp. He was supposed to travel due north for 5.7 km, but when the snow clears, he discovers that he actually traveled 7.2 km at 55o north of due east.How far (in km) must he now travel to reach base camp?

Considering as (0,0) the place where he started back to camp, he ended up at

(7.2 cos55, 7.2 sin55)(4.129,5.898),

but he wanted to end up at camp: (0,5.7)

So, the distance from camp is

d^2 = 4.129^2 + (5.898-5.7)^2
= 17.087

d = 4.133 km

To find the distance the explorer must now travel to reach the base camp, we need to use the concept of vector addition and trigonometry.

First, let's analyze the explorer's journey. He intended to travel due north for 5.7 km, but due to the whiteout, he actually traveled at an angle of 55 degrees north of due east for 7.2 km.

Now, let's break down the distance traveled into its north and east components. We'll use trigonometry to determine these components.

The north component of the distance traveled can be found by multiplying the distance (7.2 km) by the cosine of the angle (55 degrees):

North Component = 7.2 km × cos(55°)

Similarly, the east component can be found by multiplying the distance (7.2 km) by the sine of the angle (55 degrees):

East Component = 7.2 km × sin(55°)

To find the distance from the final position of the explorer to the base camp, we should calculate the difference between the north component and the northward distance initially planned.

Distance to Base Camp = Distance traveled north - North Component initially planned

The north component initially planned can be calculated using the distance initially planned (5.7 km) multiplied by the sine of 90 degrees (as the initial plan was to travel due north):

North Component initially planned = 5.7 km × sin(90°)

Finally, substitute the calculated values into the equation to find the distance to the base camp:

Distance to Base Camp = (5.7 km × sin(90°)) - (7.2 km × cos(55°))

By evaluating this equation, you can find the distance to the base camp.