Question:

A hot air balloon is rising vertically above the ground. The distance, d, between the balloon and the ground is increasing at a constant rate of 3 meters per second. At the same time, the angle of elevation, θ, between the ground and the line of sight from an observer to the balloon is decreasing at a constant rate of 0.02 radians per second.

Given that the angle of elevation from the observer to the balloon is 1 radian, find the rate at which the balloon is rising when the angle of elevation is 0.6 radians.

Answer:

To solve this problem, we need to relate the given rates of change to each other using the concept of related rates. We start by visualizing the situation and defining the variables involved:

Let d represent the distance between the balloon and the ground (in meters).

Let θ represent the angle of elevation from the observer to the balloon (in radians).

Given rates:

• rate of change of distance with respect to time, ddt = 3 m/s
• rate of change of angle of elevation with respect to time, dθdt = -0.02 rad/s

We are interested in finding the rate at which the balloon is rising when the angle of elevation is 0.6 radians, ddt when θ = 0.6.

Step 1: Determine the relationship between the variables d and θ.

Since the hot air balloon is rising vertically, we can consider a right triangle formed by the observer, the balloon, and a point on the ground directly below the balloon. The height of the balloon forms the opposite side of the triangle, while the distance from the observer to the balloon forms the hypotenuse.

Using trigonometry, we can relate the variables as follows:

tan(θ) = (height of balloon) / (distance to balloon) = d / h

Rearranging the equation, we can express the height in terms of d and θ:

height = d * tan(θ)

Step 2: Differentiate both sides of the equation with respect to time.

d(height)/dt = d(d * tan(θ))/dt

Using the chain rule on the right-hand side, we have:

d(height)/dt = d/dt (d * tan(θ)) = d(d)/dt * tan(θ) + d * d(tan(θ))/dt

Since the height is changing with respect to time, we can substitute ddt for d(height)/dt, dθdt for d(tan(θ))/dt, and d for d:

ddt = d(d)/dt * tan(θ) + d * dθdt

Step 3: Substitute given values and solve for ddt.

Putting ddt = 3 m/s, dθdt = -0.02 rad/s, and θ = 1 rad into the equation, we have:

3 = d(d)/dt * tan(1) + d * (-0.02)

3 = d(d)/dt * tan(1) - 0.02d

Step 4: Find d(d)/dt when θ = 0.6 to solve for ddt.

We are asked to find ddt when θ = 0.6. Substituting θ = 0.6 into the equation, we have:

3 = d(d)/dt * tan(0.6) - 0.02d

Now, we need to find d(d)/dt when θ = 0.6. Differentiating the equation tan(θ) = d/h implicitly with respect to time, we get:

sec^2(θ) * dθdt = d(d)/dt / h

Solving for d(d)/dt, we have:

d(d)/dt = sec^2(θ) * dθdt * h

Substituting h = d * tan(θ), we get:

d(d)/dt = sec^2(θ) * dθdt * (d * tan(θ))

Substituting θ = 0.6, we have:

d(d)/dt = sec^2(0.6) * (-0.02) * (d * tan(0.6))

Step 5: Calculate ddt when θ = 0.6.

Substituting d(d)/dt = sec^2(0.6) * (-0.02) * (d * tan(0.6)) into the equation from Step 4, we have:

3 = (sec^2(0.6) * (-0.02) * (d * tan(0.6))) * tan(0.6) - 0.02d

Simplifying the equation, we find:

3 + 0.02d = sec^2(0.6) * (-0.02) * (d * tan(0.6)) * tan(0.6)

Dividing both sides of the equation by sec^2(0.6) * (-0.02) * tan(0.6), we have:

(3 + 0.02d) / (sec^2(0.6) * (-0.02) * tan(0.6)) = d

Evaluating the right-hand side of the equation, we find:

d ≈ -10.596

Therefore, the rate at which the balloon is rising when the angle of elevation is 0.6 radians is approximately -10.596 m/s.

Final Answer: The rate at which the balloon is rising when the angle of elevation is 0.6 radians is approximately -10.596 m/s.