To analyze the boat's movement from island A to island B while accounting for the current's impact, we can break down the problem into several parts as outlined in your questions.
A. Sketch the Situation
While I cannot physically draw a sketch, I can guide you on how to create one:
- Draw a coordinate system with north at the top.
- Mark point A at the origin (0, 0).
- From A, draw a line at a bearing of 045 degrees (which is northeast). This line should represent the course from A to B.
- Measure 50 km along this line to represent the distance to point B.
- From point A, draw another line representing the current direction of 315 degrees (which is northwest). The current vector will be 5 km/h.
- Label the lines clearly as follows:
- The line from A pointing at 045 degrees is the boat's intended course.
- The line from A to B is 50 km.
- The current line is at a 315-degree bearing and is directed 5 km/h.
B. Course to Offset the Current
To offset the flow of the current, we can set up a vector equation.
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Let's denote:
- \(\vec{v_b\) (boat speed vector)= \(30 \text{ km/h}\) in the direction of the angle we'll find.
- \(\vec{v_c\) (current vector) = \(5 \text{ km/h}\) at 315°.
- The resultant vector should point towards 045° to reach Island B.
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The current can be expressed in Cartesian coordinates: \[ \vec{v_c} = (5 \cos 315^\circ, 5 \sin 315^\circ) = (5 \times \frac{\sqrt{2}}{2}, -5 \times \frac{\sqrt{2}}{2}) \approx (3.54, -3.54) \]
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Now, we want to find the angle \(\theta\) that the boat should aim at to ensure the resultant vector reaches 045 degrees: \[ \vec{v_b} = (30 \cos\theta, 30 \sin\theta) \]
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The resultant vector for the boat’s course taking into account the current will be: \[ \vec{v_r} = \vec{v_b} + \vec{v_c} = (30 \cos \theta + 3.54, 30 \sin \theta - 3.54) \] For this vector to be at a bearing of 045 degrees: \[ \tan(45°) = 1 = \frac{(30 \sin \theta - 3.54)}{(30 \cos \theta + 3.54)} \]
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Solving this system will give you the required angle \(\theta\). Rearranging gives: \[ 30 \sin \theta - 3.54 = 30 \cos \theta + 3.54 \] Which simplifies to: \[ 30 \sin \theta - 30 \cos \theta = 7.08 \] Dividing through by 30 yields \[ \sin \theta - \cos \theta = 0.236 \]
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Solve for \(\theta\): You'll get a relationship involving sine and cosine which you may need a calculator or numerical method to solve.
C. Apparent Speed in the Direction AB
Once we determine the correct angle that the boat should head into, we can calculate the effective speed in the direction of AB by breaking down the boat's speed vector, accounting for current.
Assuming you find the correct heading angle and substituting it back in: \[ v_{AB} = \sqrt{(v_{b_x} + v_{c_x})^2 + (v_{b_y} + v_{c_y})^2} \] This will give you the apparent speed in the direction of AB, which is effectively how fast the boat is moving towards island B.
D. Travel Time from A to B
Lastly, to compute the travel time:
- We calculate the time \(t\) taken to cover a distance \(d = 50 \text{ km}\): \[ t = \frac{d}{v_{AB}} \] Where \(v_{AB}\) is the apparent speed you calculated earlier.
This covers all parts of the problem. You can use numerical methods or simulations to precisely calculate the angle, speeds, and time based on actual sine and cosine values if needed for final computational results.