Final answer:
To find the distance Bear Grylls is from the camp at this point, we need to find the magnitude of the resulting displacement. Using the given values, we can calculate the final distance by breaking down the displacement into vectors, finding the x and y components, and using the Pythagorean theorem.
Step-by-step explanation:
To solve this problem, we can use vector addition. Let's break down the movements of Bear Grylls into vectors.
First, Bear Grylls solves the initial displacement by moving 20° west of north. Let's call this vector A.
We can represent vector A as A = (0, d1), where d1 is the magnitude of the displacement.
Next, Bear Grylls hikes 20° east of north for 2.25 miles, so we can represent this displacement as vector B = (d2, 20° - 90°).
To find the total displacement, we need to add vectors A and B. Since these vectors are not in the same direction, we have to use the components of the vectors to find the resulting displacement.
By using trigonometry, we can find the x and y components of vectors A and B. Then, we can add the x and y components separately to find the resulting displacement.
The x component represents the displacement in the east-west direction, and the y component represents the displacement in the north-south direction.
After finding the resulting displacement, we can use the Pythagorean theorem to find the magnitude of the displacement. The magnitude will give us the distance Bear Grylls is from the camp at this point.
Therefore, to find the distance Bear Grylls is from the camp at this point, we need to find the magnitude of the resulting displacement.
Using the given values, we can calculate the final distance:
d^2 = (d1 + d2*cos(20°))^2 + (d2*sin(20°))^2
d = √((d1 + d2*cos(20°))^2 + (d2*sin(20°))^2)
Calculating the values d1 and d2:
If the camp is 70° west of north after a displacement of d2, then we can write the equation:
d2*sin(70°) = d2*cos(20°)*cos(70°) + d2*sin(20°)*sin(70°)
By solving this equation, we can find the value of d2. Then, substitute the values of d1 and d2 into the equation for d to find the final answer.