Final Answer:
The angle opposite the northward path is changing at a rate of -25/41 radians per hour, indicating the van is slowly turning towards the city of Gainesville.
Step-by-step explanation:
Setting up the problem:
The van travels north (y-axis) and maintains its east position (x-axis) at 8 km.
We want to find the rate of change of the angle θ opposite the northward path (θ is formed between the eastward position and the van's actual direction).
Relating distance and angle:
Using the Pythagorean theorem for the right triangle formed by the van's position:
Total distance traveled = hypotenuse = √(x^2 + y^2)
When the van travels 10 km north, y = 10 km and x = 8 km.
Therefore, the total distance traveled = √(64 + 100) = √164 ≈ 12.8 km
θ = sin^-1(y/total distance) = sin^-1(10/12.8) ≈ 38.6°
Rate of change of the angle:
We need to find dθ/dt, which represents the angular velocity.
The angle changes as the van travels north, so dθ/dt will be negative since the van is turning towards the city (eastward).
dθ/dt can be calculated using the formula: dθ/dt = (dy/dt) / (total distance)
dy/dt = 20 km/h (northward speed)
Therefore, dθ/dt = (20 km/h) / (12.8 km) ≈ -1.56 rad/h
Converting to radians per hour: -1.56 rad/h * 60 min/h = -93.6 rad/h
Rounding to the nearest hundredth: -25/41 rad/h
Therefore, the angle opposite the northward path is changing at a rate of -25/41 radians per hour, indicating a slow turn towards the city of Gainesville as the van travels north.