Question

The castle of Beynac is 200 meters from the Dordogne River, as measured from the closest point P in the 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.

a) 2 meters/second
b) 4 meters/second
c) 6 meters/second
d) 8 meters/second

Answer 1

Veronica's distance to the castle is changing at a rate of 450 meters/second.

Step-by-step explanation:

To solve this problem, we can use the concept of related rates. Let's consider the position of Veronica in the canoe as a distance x from point P. We can form a right triangle with the distance from the castle to the river as the hypotenuse of the triangle, and x as one of the legs.

Using the Pythagorean theorem, we have:

x^2 + 200^2 = (450 + x)^2

Simplifying the equation, we get:

x^2 + 40000 = 202500 + 900x + x^2

Combining like terms, we have:

900x = 162500

Dividing both sides by 900, we get:

x = 162500/900 = 180.56 meters

Now, to find how fast x is changing with respect to time, we differentiate the equation with respect to t:

2x(dx/dt) = 900(dx/dt)

Dividing both sides by 2x, we get:

dx/dt = 900/2 = 450 meters/second