Question

the castle of beynac is 200 meters from the dordogne river as measured from the closest point p in center of the river. if veronica is floating down the center of the river in a canoe at a rate of 4 meters/second, how fast is her distance to the castle changing when she is 450 meters upstream from point p. you can assume that the river is straight on this stretch and ignore any issues of elevation.

Answer 1

To calculate how fast Veronica's distance to the castle is changing, we use related rates and the Pythagorean theorem to form equations and then differentiate with respect to time.

Step-by-step explanation:

To find out how fast Veronica's distance to the castle is changing, we use the concepts of related rates—a calculus method that relates the rates at which different quantities change. Given that the river is straight, we can treat this situation as a right triangle where the castle of Beynac, Veronica's canoe, and point P form the vertices. The distance from the castle to point P is one leg of the triangle (200 meters), and Veronica's distance upstream from point P is the other leg (which is changing as she moves).

Let us denote Veronica's distance from point P as x, and her distance to the castle as z. At the moment when Veronica is 450 meters upstream from point P, the problem asks us to find dz/dt (the rate at which z is changing with respect to time) when she is moving at 4 meters per second (dx/dt = -4 m/s, the negative sign indicates she is moving towards point P).

Using the Pythagorean theorem for the right triangle formed, z2 = x2 + 2002 and differentiating both sides with respect to time t, we get:

2z(dz/dt) = 2x(dx/dt)

Plugging in the values for x, dx/dt, and z (the latter found using the Pythagorean theorem: z = √(4502 + 2002)), we can solve for dz/dt, the rate at which Veronica's distance to the castle is changing.