Final answer:
The distance between the snails is changing at approximately 2.14 cm/hour after two hours based on the principles of the Pythagorean theorem and right triangle geometry.
Step-by-step explanation:
To solve for how fast the distance between the snails is changing, we need to apply the Pythagorean theorem since one snail moves north and the other east, creating a right triangle. The rate of change of the distance between the snails can be thought of as the hypotenuse of a right triangle where the legs represent the paths of the snails.
After two hours, the north-moving snail has traveled 2 hours × 10 cm/hour = 20 cm, and the east-moving snail has traveled 2 hours × 3 cm/hour = 6 cm. These distances are the legs of the right triangle, and the hypotenuse is the direct distance between the snails.
By the Pythagorean theorem, this distance is √(20² + 6²) cm = √(400 + 36) cm = √436 cm. Since the snails are moving at constant rates, the hypotenuse will also increase at a constant rate, which we can find by differentiating the hypotenuse length with respect to time.
The rate of change of the hypotenuse can be found using the relation (2 × 20 cm/h) / (√436 cm) for the north-moving snail and (2 × 6 cm/h) / (√436 cm) for the east-moving snail. Adding these rates because they are perpendicular to each other, we get the total rate of change of the hypotenuse, or the distance between the snails, which is approximately 2.14 cm/hour.