To reach a campsite, a hiker first walks for 2 mi at a bearing N 40 degree E. Then He walks 3 mi due east. What are the magnitude and direction of his hike from his stating point to the campsite? Round the distance to the nearest tenth of a mile and the direction to the nearest degree.
Draw a diagram; it helps keep the trig functions straight.
Starting from (0,0) a walk of 2 mi at N40°E leaves him at
x = 2sin40° = 1.29
y = 2cos40° = 1.53
Now, a walk 3 mi due east leaves him at
(4.29,1.53)
The new bearing is an angle θ with tanθ = 4.29/1.53 = 2.80
so, the new location is 4.55 mi N70°E
To determine the magnitude and direction of the hiker's hike from the starting point to the campsite, we can use vector addition.
First, let's convert the given bearing to a vector representation. The bearing N 40° E indicates that the direction of the hiker's first walk is 40° east of north.
Step 1: Convert the bearing to a vector representation:
To convert the bearing to a vector, we can use trigonometry. Since the bearing is east of north (N), we can create a right-angled triangle where the angle is 40°, and the side adjacent to the angle represents the northward component of the vector.
The northward component (N) can be calculated using the formula:
N = hypotenuse * cos(angle)
Given that the hiker walked 2 miles, the hypotenuse is 2.
N = 2 * cos(40°) ≈ 2 * 0.766 ≈ 1.53
So, the northward component is approximately 1.53.
Now, let's calculate the eastward component (E) using the formula:
E = hypotenuse * sin(angle)
E = 2 * sin(40°) ≈ 2 * 0.643 ≈ 1.29
So, the eastward component is approximately 1.29.
Therefore, the vector representation of the first walk is (1.53 N, 1.29 E).
Step 2: Determine the resulting vector by adding the second walk vector component-wise to the first walk vector.
The hiker walks 3 miles due east, which can be represented by the vector (0 N, 3 E).
Adding the vectors component-wise, we get:
(1.53 N, 1.29 E) + (0 N, 3 E) = (1.53 N, 4.29 E)
Step 3: Calculate the magnitude and direction of the resulting vector.
The magnitude of the resulting vector can be obtained using the Pythagorean theorem.
Magnitude = sqrt(N^2 + E^2)
Magnitude = sqrt(1.53^2 + 4.29^2)
Magnitude ≈ sqrt(2.3409 + 18.3841)
Magnitude ≈ sqrt(20.725)
Magnitude ≈ 4.6 (rounded to the nearest tenth)
The round trip distance from the starting point to the campsite is approximately 4.6 miles.
To find the direction of the resulting vector, we can use the inverse tangent function.
Direction = atan(E / N)
Direction = atan(4.29 / 1.53)
Direction ≈ atan(2.798)
Direction ≈ 70° (rounded to the nearest degree)
Therefore, the magnitude and direction of the hiker's hike from the starting point to the campsite are approximately 4.6 miles and 70°, respectively.