Final answer:
To find the equation of the tangent line to a hyperbola, you determine the slope of the tangent at the given point by either differentiating the function or using the information provided about the position or velocity at specific times.
Step-by-step explanation:
To find the equation of the tangent line to a hyperbola at a specific point, you must first determine the slope of the tangent line at that point on the curve. This can often be done by taking the derivative of the function defining the hyperbola with respect to x (if the equation is in terms of x) or with respect to t if the function is parameterized by time t.
In the scenario provided, the endpoints of the tangent at t = 25 seconds are given. These points have positions of 1300 meters at 19 seconds and 3120 meters at 32 seconds. By plugging these values into a position-time equation, you can solve for the velocity v, which is the slope of the tangent line in a velocity-time graph.
The slope can also be calculated using two known velocities at different times. For example, if you have velocities of 260 m/s and 210 m/s at times 51 s and 1.0 s, respectively, the slope a would be (260 m/s - 210 m/s) / (51 s - 1.0 s), leading to a slope of 1.0 m/s².
A quadratic equation such as at² + bt + c = 0 can arise in physics when dealing with trajectories. The constants a, b, and c in such an equation could represent physical quantities like acceleration and initial velocity, and the solutions to the quadratic equation can be used to find key points like maximum height or time to hit the ground.