Final answer:
To find the equation of the tangent line at x=a, calculate the slope based on the given endpoints and then use the point-slope form of a line equation, incorporating the specific point's coordinates.
Step-by-step explanation:
To find the equation of the tangent line at a specific point x=a, we first need to determine the slope of the tangent line. In this context, the slope of the line is given as the rate of change between two quantities. For example, if we know the change in velocity (Δv) over the change in time (Δt), then the slope (v) can be calculated using the formula:
v = Δv / Δt
Given the information, suppose we have two points on the tangent line corresponding to a position of 1,300 m at time 19 s and a position of 3,120 m at time 32 s. We then use these points to calculate the slope:
v = (3120 m - 1300 m) / (32 s - 19 s)
Once the slope is determined, we use the point-slope form of the equation of a line, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point through which the line passes. In this case, the 'y1' and 'x1' would correspond to the values of position and time at x=a. If, for example, a is 25 s, and we found the position at that time, we'd plug those values into the point-slope form to get the final equation of the tangent line.