Bear Grylls is approximately 2.83 miles away from the camp at this point.
To determine the distance of Bear Grylls from the camp, we need to use vector addition.
Let's break down his movements into two components: north and west. Initially, Bear Grylls spots the camp 30 degrees west of north.
Then he hikes east of north for 3 miles and determines that the camp is now 65 degrees west of north.
We'll consider the north component as positive and the west component as negative.
To find the north component of the camp's location, we use the sine function. sin(30) = north component / distance from the camp.
Rearranging the formula, we have north component = sin(30) * distance from the camp.
Similarly, to find the west component, we use the cosine function. cos(30) = west component / distance from the camp.
Rearranging the formula, we have west component = cos(30) * distance from the camp.
By adding the north and west components, we can find the distance from the camp at this point: distance from the camp = sqrt(north component^2 + west component^2).
Substituting the values, we get: distance from the camp = sqrt((sin(30) * distance from the camp)^2 + (cos(30) * distance from the camp - 3)^2).
Solving this equation yields the distance from the camp as approximately 2.828 miles or 2.83 miles when rounded to the nearest hundredth.