Final answer:
To find the equation of the tangent line, determine the endpoints of the tangent from the figure, calculate the slope using the change in position over time, and then use a known point and the slope to construct the equation in point-slope form.
Step-by-step explanation:
To determine the equation of a tangent line to a curve at a given point, you need to know the slope of the tangent line at that point and a point through which the line passes. In the given problem, we are trying to find the tangent line to the curve at t = 25 s.
- First, determine the endpoints of the tangent line which corresponds to the curve's position at two different times. In this case, the positions are 1300 m at 19 s and 3120 m at 32 s.
- Next, calculate the slope (v) of the tangent line using the slope formula Δy/Δx, which in this context is the difference in position over the difference in time.
If you have the graph available, you can directly determine endpoints from the figure to find the slope, which is represented as follows:
(260 m/s - 210 m/s) / (51 s - 1.0 s) = 1.0 m/s².
Once you have the slope and a point, you can use them to construct the equation of the tangent line using the point-slope form of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point.