6. A plane is flying horizontally at a constant height of 4000 ft above a fixed observation point. At a certain instant the angle of elevation is $$ \pi / 6 $$ and decreasing, and the speed of the plane is 300 mph . How fast is the angle decreasing?

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To determine how fast the angle of elevation $$ \theta $$ is decreasing, let's analyze the given scenario step by step.

### Given:
- **Height of the plane ($$ h $$)**: 4000 ft
- **Angle of elevation ($$ \theta $$)**: $$ \frac{\pi}{6} $$ radians
- **Speed of the plane ($$ v $$)**: 300 mph

### Objective:
Find the rate at which the angle $$ \theta $$ is decreasing, denoted as $$ \frac{d\theta}{dt} $$.

### Step 1: Relate the Angle to Distance
The plane is flying horizontally, so the horizontal distance $$ x $$ from the observation point to the plane can be related to the angle $$ \theta $$ using the tangent function:
$$
\tan(\theta) = \frac{h}{x}
$$
$$
x = \frac{h}{\tan(\theta)} = h \cot(\theta)
$$

### Step 2: Differentiate with Respect to Time
Differentiate both sides of the equation with respect to time $$ t $$:
$$
\frac{dx}{dt} = -h \csc^2(\theta) \frac{d\theta}{dt}
$$
Where $$ \csc(\theta) = \frac{1}{\sin(\theta)} $$.

### Step 3: Convert Plane's Speed to Consistent Units
Convert the plane's speed from miles per hour (mph) to feet per minute (ft/min):
$$
300 \text{ mph} = 300 \times 5280 \text{ ft/mile} \div 60 \text{ min/hr} = 26400 \text{ ft/min}
$$
So, $$ \frac{dx}{dt} = 26400 \text{ ft/min} $$.

### Step 4: Solve for $$ \frac{d\theta}{dt} $$
Plugging the known values into the differentiated equation:
$$
26400 = -4000 \times \csc^2\left(\frac{\pi}{6}\right) \times \frac{d\theta}{dt}
$$
$$
\csc\left(\frac{\pi}{6}\right) = 2 \Rightarrow \csc^2\left(\frac{\pi}{6}\right) = 4
$$
$$
26400 = -4000 \times 4 \times \frac{d\theta}{dt}
$$
$$
\frac{d\theta}{dt} = -\frac{26400}{16000} = -1.65 \text{ radians per minute}
$$

### Conclusion
The angle of elevation $$ \theta $$ is decreasing at a rate of **1.65 radians per minute**.

**Answer:**  
After simplifying, the angle is decreasing at –1.65 radians per minute
The angle of elevation is decreasing at a rate of –1.65 radians per minute.

A plane is flying horizontally at a constant height of 4000 ft above a fixed observation point. At a certain
instant the angle of elevation is and decreasing, and the speed of the plane is 300 mph . How fast is the
angle decreasing?

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A plane is flying horizontally at a constant height of 4000 ft above a fixed observation point. At a certain instant the angle of elevation is and decreasing, and the speed of the plane is 300 mph . How fast is the angle decreasing?