Final answer:
The equation of a tangent plane involves calculating the slope at a given point on a surface. For motion over time, the slope represents velocity, which can be found using endpoint positions and times from a curve.
Step-by-step explanation:
To find the equation of the tangent plane to a surface at a given point, one must first determine the gradient of the surface at that point. The gradient vector will be perpendicular to the tangent plane. Given a point Q on a curve representing the motion of an object over time, we can determine the slope of the tangent line at that point, which represents the velocity of the object at time t = 25 s.
The endpoints of the tangent line can be determined from the curve. They correspond to specific positions and times. In the question, these are given as positions at 1,300 meters at 19 seconds and 3,120 meters at 32 seconds. The slope of the tangent line, or velocity v, can be found by using the differences in these positions and times.
To solve for the slope v, you can use the formula for the slope of a straight line, which is the change in position over the change in time. In this case, the slope a can be calculated as (260 m/s - 210 m/s) / (51 s - 1.0 s) = 1.0 m/s².