Final answer:
To calculate the slope of a tangent line, identify the point of tangency, find two endpoints corresponding to known positions and times, and apply the slope formula to get the velocity at that point, yielding a result of approximately 140 m/s for the given example.
Step-by-step explanation:
How to Calculate the Slope of a Tangent Line at a Given Point
To compute the slope of a tangent line to a curve at a specific point, follow these steps:
- Identify the point on the curve where the tangent line intersects. In this case, it's at t = 25 s.
- Determine the endpoints of the tangent that correspond to specific positions and times. According to the information provided, one endpoint corresponds to a position of 1300 m at time 19 s, and the other corresponds to a position of 3120 m at time 32 s.
- Using the two positions and times, plug them into the slope formula to calculate the slope (v). The slope can be calculated as (3120 m - 1300 m) / (32 s - 19 s), which equates to 1820 m / 13 s or approximately 140 m/s.
Finally, the slope can be interpreted as the velocity of the moving object at the point t = 25 s. The slope or velocity is the change in position over time, reflecting how fast the object is moving along the curve at that point.