Q. 27 - Calculus

Question

Graph the equations in Exercises 25–32 in the polar plane. Compare your graphs with the corresponding graphs in Exercises 17–24. 

$r=2+\sin \theta$

Step-by-Step Solution

Answer

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Step 1. Given information

The equation of the curve:

$r=2+\sin \theta$ 

Step 2: Draw a diagram and identify given information

Draw a right triangle or appropriate diagram based on the problem description. Label the known angle and the given speed/distance. Identify which component (horizontal or vertical) corresponds to the quantity being asked for.

Step 3: Set up the trigonometric equation

Using the right triangle, the vertical component (altitude gain) is found using \(\text{vertical} = v \sin(\theta)\) and the horizontal component is found using \(\text{horizontal} = v \cos(\theta)\), where \(v\) is the speed and \(\theta\) is the angle of elevation.

Step 4: Substitute values and compute

Substitute the given numerical values into the trigonometric equation and compute the result. Make sure to use the correct units throughout the calculation.

Step 5: State the final answer

Express the final answer with appropriate units and in the context of the original problem.