Final answer:
To find the equation of the tangent line at t = 25 s for a parametric curve, the endpoints of the tangent are determined, and the slope is calculated using the formula v = (Δy/Δt), with Δy and Δt representing the changes in position and time respectively.
Step-by-step explanation:
To find the equation of a tangent line to a curve at the point determined by a specified value of t, we must first understand that the slope of a curve at a point is equal to the slope of the straight line tangent to the curve at that point. The procedure for finding the tangent line at t = 25 s involves several steps which utilize the concept of slope (v) and endpoint coordinates.
- Find the tangent line to the curve at t = 25 s.
- Determine the endpoints of the tangent, which correspond to known positions and times.
- Use these endpoints to calculate the slope (v) by plugging them into the equation that represents the slope, which is the change in position over the change in time.
For example, if the endpoints are at (19 s, 1300 m) and (32 s, 3120 m), we can use these to solve for the slope by applying the formula v = (Δy/Δt), where Δy is the change in position and Δt is the change in time.