Logo

AskSia

Plus

Current Attempt in Progress Your answer is partially correct. Oasis B is $9.0 ...
Jun 18, 2024
Solution
a
First, let's break down the camel's journey into components. The camel walks 23 km23 \, \text{km} in a direction 15.0∘15.0^\circ south of east. We can find the x and y components of this displacement: x1=23cos⁡(15∘)y1=−23sin⁡(15∘) x_1 = 23 \cos(15^\circ) \\ y_1 = -23 \sin(15^\circ) Calculating these values: x1=23cos⁡(15∘)≈22.2 kmy1=−23sin⁡(15∘)≈−5.96 km x_1 = 23 \cos(15^\circ) \approx 22.2 \, \text{km} \\ y_1 = -23 \sin(15^\circ) \approx -5.96 \, \text{km}
b
Next, the camel walks 34 km34 \, \text{km} due north. This only affects the y-component: x2=0y2=34 km x_2 = 0 \\ y_2 = 34 \, \text{km}
c
Now, we sum the components to find the total displacement from oasis A: xtotal=x1+x2=22.2+0=22.2 kmytotal=y1+y2=−5.96+34=28.04 km x_{\text{total}} = x_1 + x_2 = 22.2 + 0 = 22.2 \, \text{km} \\ y_{\text{total}} = y_1 + y_2 = -5.96 + 34 = 28.04 \, \text{km}
d
The camel needs to walk directly to oasis B, which is 9.0 km9.0 \, \text{km} due east of oasis A. The coordinates of oasis B relative to A are: xB=9.0 kmyB=0 km x_B = 9.0 \, \text{km} \\ y_B = 0 \, \text{km} The displacement vector from the camel's current position to oasis B is: Δx=xB−xtotal=9.0−22.2=−13.2 kmΔy=yB−ytotal=0−28.04=−28.04 km \Delta x = x_B - x_{\text{total}} = 9.0 - 22.2 = -13.2 \, \text{km} \\ \Delta y = y_B - y_{\text{total}} = 0 - 28.04 = -28.04 \, \text{km}
e
The distance the camel needs to walk is the magnitude of this displacement vector: d=(Δx)2+(Δy)2=(−13.2)2+(−28.04)2≈31.1 km d = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(-13.2)^2 + (-28.04)^2} \approx 31.1 \, \text{km}
f
The direction relative to the positive x-axis is given by the angle: θ=tan⁡−1(ΔyΔx)=tan⁡−1(−28.04−13.2)≈64.8∘ \theta = \tan^{-1}\left(\frac{\Delta y}{\Delta x}\right) = \tan^{-1}\left(\frac{-28.04}{-13.2}\right) \approx 64.8^\circ Since both Δx\Delta x and Δy\Delta y are negative, the angle is in the third quadrant: θ=180∘−64.8∘=−115.2∘ \theta = 180^\circ - 64.8^\circ = -115.2^\circ
Answer
The camel should walk approximately 31.1 km31.1 \, \text{km} at an angle of −115.2∘-115.2^\circ relative to the positive x-axis.
Key Concept
Vector addition and trigonometry are used to determine the resultant displacement and direction.
Explanation
By breaking down the camel's journey into x and y components, summing these components, and then calculating the resultant vector, we can determine the distance and direction the camel needs to walk to reach oasis B.
© 2023 AskSia.AI all rights reserved