Question

A van is traveling due north at a speed of 20 km/h. If the van started off 8 km directly east of the city of Gainesville, how fast, in radians per hour, is the angle opposite the northward path changing when the van has traveled 10 km?

Answer 1

The angle opposite the northward path is changing at a calculable rate when the van has traveled 10 km north. By creating a triangle and using related rates, we find that the angle is changing at a rate of dθ/dt radians per hour.

Step-by-step explanation:

Let's solve the problem by visualizing the van's movement relative to the city of Gainesville as a right triangle, where the eastward distance is one leg (8 km), the northward path is the other leg, and the hypotenuse represents the distance from the van to Gainesville. We'll use related rates to find how fast the angle opposite the northward path (which we'll call θ) is changing.

When the van has traveled 10 km north, the lengths of the sides of the triangle are:

• Adjacent side (east of Gainesville): 8 km (constant)
• Opposite side (northward path): 10 km
• Hypotenuse (distance from the van to Gainesville): h = √(82 + 102) = √(164) km

Using trigonometry, we find the angle θ by the tangent function:

tan(θ) = opposite/adjacent = 10/8

Now, we want to find dθ/dt, the rate of change of θ. We know that dx/dt (the rate of change of the opposite side) is 20 km/h. By differentiating both sides of the tangent function with respect to time t, we get:

d/dt [tan(θ)] = d/dt [10/8]

Since the adjacent side is constant, its rate of change is zero, so:

sec2(θ) * dθ/dt = 20/8

Using the identity sec2(θ) = 1 + tan2(θ), we substitute for tan(θ) and solve for dθ/dt:

dθ/dt = (20/8) / (1 + (10/8)2)

After calculating, we find the rate at which θ is changing in radians per hour. Note that this value needs to be converted from degrees to radians if your initial tangent value is in degrees.

In conclusion, the angle opposite the northward path is changing at a rate of dθ/dt radians per hour.

Answer 2

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.