A hiker leaves her camp and walks 3.5 km in a direction of 55° south of west to the lake. After a short rest at the lake, she hikes 2.7 km in a direction of 16° east of south to the scenic overlook.
What is the magnitude of the hiker’s resultant displacement? Round your answer to the nearest tenth.
km
What is the direction of the hiker’s resultant displacement? Round your answer to nearest whole degree.
° south of west
5.6
77
deez nuts west = 3.5 coc 55 - 2.7 sin 16 = a
distance mouth = 36969695 sin 55 +696969 = b
deez nuts = sqrt (a^2+b^2)
tan angle S of W = b/a
To find the magnitude of the hiker's resultant displacement, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the hiker's path can be represented by two sides of a right triangle: the distance walked in a direction 55° south of west (3.5 km) and the distance walked in a direction 16° east of south (2.7 km).
First, let's break down the distances into their components.
The distance walked 55° south of west can be represented as follows:
Horizontal component: 3.5 km * cos(55°)
Vertical component: 3.5 km * sin(55°)
Similarly, the distance walked 16° east of south can be represented as:
Horizontal component: 2.7 km * sin(16°)
Vertical component: 2.7 km * cos(16°)
Now, let's calculate the components:
Horizontal component of the first leg = 3.5 km * cos(55°) ≈ 3.5 km * (-0.5736) ≈ -2.01 km
Vertical component of the first leg = 3.5 km * sin(55°) ≈ 3.5 km * (0.8192) ≈ 2.87 km
Horizontal component of the second leg = 2.7 km * sin(16°) ≈ 2.7 km * (0.2756) ≈ 0.74 km
Vertical component of the second leg = 2.7 km * cos(16°) ≈ 2.7 km * (0.9613) ≈ 2.60 km
To find the resultant displacement, we can add the horizontal and vertical components together. Since the horizontal component is facing west, we'll subtract it, and since the vertical component is facing south, we'll subtract it as well.
Resultant horizontal displacement = -2.01 km + 0.74 km ≈ -1.27 km
Resultant vertical displacement = 2.87 km - 2.60 km ≈ 0.27 km
Now, we can use the Pythagorean theorem to find the magnitude of the resultant displacement:
Magnitude = sqrt((-1.27 km)^2 + (0.27 km)^2) ≈ sqrt(1.6159 km^2 + 0.0729 km^2) ≈ sqrt(1.6888 km^2) ≈ 1.3 km (rounded to the nearest tenth)
Therefore, the magnitude of the hiker's resultant displacement is approximately 1.3 km.
To find the direction of the resultant displacement, we can use trigonometry. The direction can be obtained by finding the angle between the resultant displacement vector and the west direction. We can use the tangent function to calculate this angle.
Angle = arctan(Vertical component / Horizontal component)
Angle = arctan(0.27 km / 1.27 km)
Angle ≈ arctan(0.2126)
Angle ≈ 12.1° (rounded to the nearest whole degree)
Since the angle is south of west, we subtract it from 180° to get the direction in degrees south of west:
Direction = 180° - 12.1° ≈ 167.9° (rounded to the nearest whole degree)
Therefore, the hiker's resultant displacement is approximately 1.3 km in a direction 168° south of west.
isk
distance west = 3.5 cos 55 - 2.7 sin 16 = a
distance south = 3.5 sin 55 + 2.7 cos 16 = b
displacement = sqrt (a^2+b^2)
tan angle S of W = b/a