Oasis B is a distance d = 7.0 km east of oasis A, along the x axis shown in the figure. A confused camel, intending to walk directly from A to B instead walks a distance W1 = 22 km west of due south by angle θ1 = 15.0°. It then walks a distance W2 = 34 km due north. If it is to then walk directly to B, (a) how far (in km) and (b) in what direction should it walk (relative to the positive direction of the x axis within the range (-180°, 180°])?
Note: Xt = 7-(-3.82) = 10.82 km.
To find the distance and direction the camel should walk from its current position to reach oasis B, we can break down the problem into two components: horizontal and vertical.
(a) Finding the horizontal distance:
Given that oasis B is 7.0 km east of oasis A, and the camel has already walked 22 km west of due south, we can calculate the horizontal distance it still needs to cover.
Horizontal distance = Distance from A to B - Horizontal distance covered by the camel
Horizontal distance = 7.0 km - 22 km * cos(θ1)
Using the provided value of θ1 = 15.0°:
Horizontal distance = 7.0 km - 22 km * cos(15.0°)
(b) Finding the vertical distance:
The camel has already walked 34 km due north, so the vertical distance it still needs to cover is simply the difference between the y-coordinate of the current position and B.
Vertical distance = Distance from A to B - Vertical distance covered by the camel
Vertical distance = 0 km - 34 km
Now, we have the horizontal and vertical distances. Using trigonometry, we can calculate the total distance and direction the camel should walk.
Total distance = √(Horizontal distance² + Vertical distance²)
Direction = arctan(Vertical distance / Horizontal distance)
Substituting the calculated values, we can get the final answer.
To determine the distance and direction the camel should walk to reach oasis B, we can break down the camel's journey into several steps.
Step 1: Calculate the displacement of the camel due to its initial westward and southward deviation.
First, we need to find the x and y components of the camel's displacement caused by walking 22 km west of due south by an angle of 15.0°.
The x-component, denoted as Δx1, can be found by using the cosine function:
Δx1 = W1 * cos(θ1)
Substituting the given values:
Δx1 = 22 km * cos(15.0°)
Using a calculator, evaluate this expression:
Δx1 ≈ 21.389 km (rounded to three decimal places)
The y-component, denoted as Δy1, can be found by using the sine function:
Δy1 = W1 * sin(θ1)
Substituting the given values:
Δy1 = 22 km * sin(15.0°)
Using a calculator, evaluate this expression:
Δy1 ≈ 5.671 km (rounded to three decimal places)
Step 2: Calculate the final displacement of the camel after walking north.
After walking north a distance of 34 km, the camel's y-component displacement will increase by 34 km. Since it moves only along the y-axis, there will be no change in the x-component displacement.
The final x-component displacement, denoted as Δx2, remains the same:
Δx2 = Δx1 ≈ 21.389 km
The final y-component displacement, denoted as Δy2, will be:
Δy2 = Δy1 + 34 km
Substituting the known values:
Δy2 ≈ 5.671 km + 34 km
Evaluate this expression:
Δy2 ≈ 39.671 km (rounded to three decimal places)
Step 3: Calculate the distance and direction to reach oasis B.
The distance from oasis A to B is the magnitude of the displacement vector traveled by the camel.
The distance, d_B, can be found using the Pythagorean theorem:
d_B = sqrt(Δx2^2 + Δy2^2)
Plugging in the computed values:
d_B = sqrt((21.389 km)^2 + (39.671 km)^2)
Evaluate this expression:
d_B ≈ 45.012 km (rounded to three decimal places)
The direction of the camel's path, denoted as θ2, can be calculated using the inverse tangent function:
θ2 = arctan(Δy2 / Δx2)
Plugging in the computed values:
θ2 = arctan(39.671 km / 21.389 km)
Evaluate this expression (taking into account the correct quadrant):
θ2 ≈ 61.074° (rounded to three decimal places)
Since the direction is relative to the positive direction of the x-axis, the angle is measured counterclockwise from the positive x-axis. Therefore, the camel should walk approximately 45.012 km at a direction of around 61.074° relative to the positive x-axis to reach oasis B.
a. 22km[260o] + 34[90o].
X = 22*cos260 = -3.82 km.
Xt = 7.0 (-3.820 = 10.82 km = Total hor
distance to B.
Y = 22*sin260 + 34*sin90 = 12.33 km.
D = sqrt((11.82)^2 + 12.33^2)= 17.1 km
b. Tan A = Y/Xt = 12.33/10.82 = 1.13956
A = 48.7o