Final answer:
To find the first coordinate of the intersection point of two tangent lines, calculate the slope of each tangent at given points, write the equations of both tangents, and then solve for the x-coordinate when the equations are set equal to each other.
Step-by-step explanation:
Finding the First Coordinate of the Intersection Point of Two Tangent Lines
The task involves finding the first coordinate (usually the x-coordinate) of the intersection point of two tangent lines to a curve. Here's a methodical approach to solve this problem using calculus and algebra:
- Find the tangent line to the curve at t = 25 s. This requires knowing the function representing the curve and differentiating it to find the slope at the desired point.
- Determine the endpoints of the tangent. As given, the endpoints at t = 19 s correspond to a position of 1300 m, and at t = 32 s to a position of 3120 m.
- Using these endpoints, calculate the slope v of the tangent line by applying the slope formula Δy/Δx, which represents the change in position over the change in time.
- Find the equation of the tangent line using the point-slope form with the calculated slope and one of the endpoints.
- Repeat the process for the second tangent line if its endpoints are known, or use a similar method to find its equation.
- Finally, equate the two tangent line equations and solve for the x-coordinate to find the first coordinate of the intersection point.
It may also be helpful to sketch diagrams if the problem involves understanding substitution and income effects, ensuring that all points of tangency are correctly identified.