Final answer:
The slope of the tangent line to a polar curve involves converting to rectangular coordinates, finding the angle of inclination, determining the radius at a specific angle, and calculating the arc length using calculus.
Step-by-step explanation:
To find the slope of the tangent line to a polar curve at a given angle, you can employ several mathematical techniques. Firstly, convert the polar coordinates to rectangular coordinates since it is simpler to work with Cartesian equations when calculating slopes and tangents. To calculate the slope or the angle of inclination, you can use the formula for the derivative of the polar equation with respect to θ. The radius at a specific angle can be determined directly from the polar equation by substituting the angle value into the equation. Lastly, the arc length of a curve over a specific interval can be calculated using calculus by integrating the length element derived from the polar equation.
An example of determining the slope of the line tangent to a curve is by finding the endpoints of the tangent. These endpoints could correspond to specific times and positions, for instance, a position of 1,300 m at time 19 s and a position of 3,120 m at time 32 s. The slope, denoted as v, can then be calculated by using the 'rise over run' method with these two points.