pre calculus vector unit question: A boat is travelling at a speed of
\[20\,\dfrac{\text{km}}{\text{h}}\] in a direction that is a
\[210\degree\] rotation from east.
At a certain point it encounters a current at a speed of
\[12\,\dfrac{\text{km}}{\text{h}}\] in a direction that is a
\[40\degree\] 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.
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We have a boat traveling at 20 kilometers per hour in a direction 210 degrees from east. The boat encounters a current moving at 12 kilometers per hour at 40 degrees from east. We need to find the boat's final speed by adding these two velocity vectors together.
To add vectors, we need to convert them from polar to rectangular coordinates. We use the formulas v-x equals magnitude times cosine theta, and v-y equals magnitude times sine theta. For the boat velocity: x-component is 20 times cosine 210 degrees equals negative 17.32, y-component is 20 times sine 210 degrees equals negative 10.00. For the current: x-component is 12 times cosine 40 degrees equals 9.19, y-component is 12 times sine 40 degrees equals 7.71.
To add vectors, we add their corresponding components. The resultant x-component equals the boat's x-component plus the current's x-component: negative 17.32 plus 9.19 equals negative 8.13. The resultant y-component equals negative 10.00 plus 7.71 equals negative 2.29. We visualize this by placing the current vector at the tip of the boat vector, forming a vector triangle where the resultant connects the origin to the final tip.
To find the magnitude of the resultant vector, we use the Pythagorean theorem. The magnitude equals the square root of the sum of the squared components. Substituting our values: magnitude equals square root of negative 8.13 squared plus negative 2.29 squared, which equals square root of 66.10 plus 5.24, equals square root of 71.34, which gives us 8.4 kilometers per hour. Therefore, the boat's final speed after encountering the current is 8.4 kilometers per hour.