Final answer:
To find the tangent line to a curve at a certain point, calculate the slope at that point and use it to form parametric equations for the line. The slope is obtained by taking the change in position over the change in time. The parametric equations consider the components of the slope and the initial point of tangency.
Step-by-step explanation:
To find the parametric equations for the line that is tangent to the curve at a given point, we must first find the slope of the tangent at the specified time (t = 25 seconds). Using the positions given at two times, we can calculate the slope v of the tangent line by using the change in position over the change in time. This can be calculated using the formula:
slope, v = \( \frac{{position_{32s} - position_{19s}}}{{time_{32s} - time_{19s}}} \) = \( \frac{{3120 m - 1300 m}}{{32 s - 19 s}} \).
After calculating the slope, we can write the parametric equations x(t) and y(t), assuming the curve is in two dimensions and the motion is linear. The equations will be:
x(t) = x_0 + v_x \cdot (t - t_0)
y(t) = y_0 + v_y \cdot (t - t_0)
where x_0 and y_0 are the coordinates at the starting point of the tangent line, v_x and v_y are the components of the slope at t = 25 seconds, and t_0 is the initial time.