Final answer:
To find how fast the distance between two buses changes after half an hour, use the Pythagorean theorem and related rates. Calculate the distances each bus has traveled and differentiate it with respect to time to find the rate of change.
Step-by-step explanation:
The question involves calculating how fast the distance between two vehicles is changing. Since the vehicles are departing at right angles from the same point, we can approach this problem using the Pythagorean theorem in a related rates context.
Step-by-Step Solution
Let the distance traveled by the bus heading north be y (in miles) and the one heading east be x (in miles)
After half an hour (0.5 hours), we know y = 0.5 × 66 mph and x = 0.5 × 74 mph.Compute the distances: y = 33 miles and x = 37 miles.Use the Pythagorean theorem to express the distance between the buses d: d² = x² + y².Substitute x and y to find the distance at the half-hour mark.Differentiate both sides of the equation with respect to time t to get 2d · (dd/dt) = 2x · (dx/dt) + 2y · (dy/dt).Substitute x, y, dx/dt, and dy/dt (being the speeds of the buses) into the differentiated equation to find dd/dt, the rate at which the distance is changing.Solve for dd/dt to obtain the answer.
Finding the Changing Rate of Distance
Applying the differentiation gives us 2 ×(33² + 37²)^(1/2) × (dd/dt) = 2 × 37 × 74 + 2 × 33 × 66, from which we can solve for dd/dt. The rate at which the distance is changing after half an hour turns out to be the speed of separation between the two buses.