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A boat is travelling at a speed of 20(km)/(h) in a direction that is a 210^(@) rotation from east. At a certain point it encounters a current at a speed of 12(km)/(h) in a direction that is a 40^(@) rotation from east. What is the boat's speed after it meets the current? Round your answer to the nearest tenth. You can round intermediate values to the nearest hundredth. (km)/(h)

A boat is travelling at a speed of 20kmh 20 \frac{\mathrm{km}}{\mathrm{h}} in a direction that is a 210∘ 210^{\circ} rotation from east.\newlineAt a certain point it encounters a current at a speed of 12kmh 12 \frac{\mathrm{km}}{\mathrm{h}} in a direction that is a 40∘ 40^{\circ} rotation from east.\newlineWhat is the boat's speed after it meets the current?\newlineRound your answer to the nearest tenth. You can round intermediate values to the nearest hundredth.\newlinekmh \frac{\mathrm{km}}{\mathrm{h}}

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Q. A boat is travelling at a speed of 20kmh 20 \frac{\mathrm{km}}{\mathrm{h}} in a direction that is a 210∘ 210^{\circ} rotation from east.\newlineAt a certain point it encounters a current at a speed of 12kmh 12 \frac{\mathrm{km}}{\mathrm{h}} in a direction that is a 40∘ 40^{\circ} rotation from east.\newlineWhat is the boat's speed after it meets the current?\newlineRound your answer to the nearest tenth. You can round intermediate values to the nearest hundredth.\newlinekmh \frac{\mathrm{km}}{\mathrm{h}}
  1. Identify Vectors: Identify the vectors representing the boat's speed and the current's speed. The boat's speed vector is 20 km/h20\,\text{km/h} in a direction 210210 degrees from east, and the current's speed vector is 12 km/h12\,\text{km/h} in a direction 4040 degrees from east. We will use vector addition to find the resultant speed of the boat after it meets the current.
  2. Break Down Components: Break down the vectors into their components.\newlineThe boat's speed components:\newline- x-component (East-West axis): 20×cos⁡(210°)20 \times \cos(210°)\newline- y-component (North-South axis): 20×sin⁡(210°)20 \times \sin(210°)\newlineThe current's speed components:\newline- x-component (East-West axis): 12×cos⁡(40°)12 \times \cos(40°)\newline- y-component (North-South axis): 12×sin⁡(40°)12 \times \sin(40°)
  3. Calculate Boat's Speed: Calculate the components of the boat's speed.\newlineBoat's x-component: 20×cos⁡(210°)=20×(−0.866)≈−17.3220 \times \cos(210°) = 20 \times (-0.866) \approx -17.32 km/h (West)\newlineBoat's y-component: 20×sin⁡(210°)=20×(−0.5)=−1020 \times \sin(210°) = 20 \times (-0.5) = -10 km/h (South)
  4. Calculate Current's Speed: Calculate the components of the current's speed.\newlineCurrent's x-component: 12×cos⁡(40°)≈12×0.766≈9.19 km/h (East)12 \times \cos(40°) \approx 12 \times 0.766 \approx 9.19 \text{ km/h} \text{ (East)}\newlineCurrent's y-component: 12×sin⁡(40°)≈12×0.643≈7.72 km/h (North)12 \times \sin(40°) \approx 12 \times 0.643 \approx 7.72 \text{ km/h} \text{ (North)}
  5. Add Vector Components: Add the components of the boat's speed and the current's speed to find the resultant vector components.\newlineResultant x-component: −17.32 km/h-17.32 \, \text{km/h} (boat) + 9.19 km/h9.19 \, \text{km/h} (current) ≈−8.13 km/h\approx -8.13 \, \text{km/h} (West)\newlineResultant y-component: −10 km/h-10 \, \text{km/h} (boat) + 7.72 km/h7.72 \, \text{km/h} (current) ≈−2.28 km/h\approx -2.28 \, \text{km/h} (South)
  6. Calculate Resultant Speed: Calculate the magnitude of the resultant vector to find the boat's resultant speed.\newlineResultant speed = ((−8.13)2+(−2.28)2)≈(66.11+5.19)≈(71.30)≈8.44 km/h\sqrt{((-8.13)^2 + (-2.28)^2)} \approx \sqrt{(66.11 + 5.19)} \approx \sqrt{(71.30)} \approx 8.44 \, \text{km/h}
  7. Round Resultant Speed: Round the resultant speed to the nearest tenth.\newlineResultant speed ≈8.4\approx 8.4 km/h

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