Final answer:
To find the tangent line's equation at t = 25 s, determine the slope using the positions at t = 19 s and t = 32 s, then apply the point-slope form of a line with this slope and a point on the curve.
Step-by-step explanation:
Finding the equation of the tangent line to a curve at a specific point involves determining the slope of the curve at that point. According to the given problem, the student needs to find this equation at the point corresponding to t = 25 seconds (s). We are given that the endpoints of the tangent line at t = 19 s correspond to a position of 1300 meters (m) and at t = 32 s to a position of 3120 m.
To calculate the slope (v), we use the difference in position divided by the difference in time. For the mentioned positions and times, the slope would be (3120 m - 1300 m) / (32 s - 19 s) which simplifies to 1820 m / 13 s or approximately 140 m/s.
After finding the slope, we would use the point-slope form of a line to write the equation of the tangent line. The point-slope form is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point on the curve. In the context of this problem, x would correspond to time t, and y would correspond to position.