Final answer:
To find the parametric equations for a curve, calculate the slope (velocity) using two points on a tangent line. The independent variable is time, and the dependent variable is position. The slope describes the rate of change of position over time.
Step-by-step explanation:
When given the endpoints of a tangent line and a specific point in time, one can find the parametric equations for the curve by first determining the slope of the tangent, which represents the instantaneous velocity at that point. Considering the positions at time 19 s and 32 s, we use the formula for slope (v = (y2 - y1) / (t2 - t1)) to find the slope, with y representing position in meters and t representing time in seconds.
The y-intercept is the position at which the tangent line would cross the y-axis, which correlates to the time being equal to zero. Since the problem does not provide enough data to calculate this directly, one would typically extend the tangent line backward, using the slope and a point it passes through, to find the y-intercept.
The independent variable in this scenario is time (t), while the dependent variable is position (y). The slope (v) represents the rate of change of position over time, an essential concept in understanding motion under uniform velocity.