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An airplane is flying level at an altitude of 3000 feet and a speed of 200 ft/sec. It is about to pass over a radar station on the ground, as shown below. The dashed line represents the line-of-sight from an operator in the radar station to the airplane.
plane
How fast is the distance between the station and the airplane changing at the instant when the plane is descending to x = 2000 feet? If you are unsure how to proceed, follow the steps listed below.
Step 1. You must create a new variable. Click here for tips.
Step 2. Write a one sentence description of what your new variable measures. Get feedback on this if possible.
Step 3. Create an equation that involves your new variable.
Step 4. Solve the problem. It may help to organize your variables and data. Be accurate to one decimal place and include units.

This is a related rates problem involving the distance between the airplane and the radar station. The airplane flies level at a constant altitude of 3000 ft and a constant horizontal speed of 200 ft/sec, and we are tasked with finding how fast the distance between the radar station and the airplane is changing when \( x = 2000 \) feet.

### Step-by-Step Solution:

#### 1. **Introduce Variables**:
- Let \( s \) represent the distance between the radar station and the airplane.
- Let \( x \) represent the horizontal distance between the radar station and the airplane.
- The altitude of the airplane is constant at 3000 feet, so we denote this as a constant \( h = 3000 \, \text{ft} \).

#### 2. **Set up the relation between the variables**:
Using the Pythagorean theorem, we relate \( s \), \( x \), and \( h \) (which is constant):
\[
s^2 = x^2 + h^2
\]
Since \( h = 3000 \, \text{ft} \), this simplifies to:
\[
s^2 = x^2 + 3000^2
\]
Thus, the distance between the airplane and the radar station is given by:
\[
s = \sqrt{x^2 + 3000^2}
\]

#### 3. **Differentiate the equation with respect to time**:
Now, we differentiate both sides of the equation with respect to time \( t \) to find the rate of change of \( s \) (denoted \( \frac{ds}{dt} \)) with respect to time:
\[
2s \frac{ds}{dt} = 2x \frac{dx}{dt}
\]
Simplifying:
\[
s \frac{ds}{dt} = x \frac{dx}{dt}
\]
Solve for \( \frac{ds}{dt} \):
\[
\frac{ds}{dt} = \frac{x}{s} \cdot \frac{dx}{dt}
\]

#### 4. **Substitute Known Values**:
- At the instant of interest, \( x = 2000 \, \text{ft} \).
- The speed of the airplane (horizontal velocity) is \( \frac{dx}{dt} = 200 \, \text{ft/sec} \).
- Now calculate \( s \) when \( x = 2000 \, \text{ft} \):
\[
s = \sqrt{2000^2 + 3000^2} = \sqrt{4000000 + 9000000} = \sqrt{13000000} \approx 3605.6 \, \text{ft}
\]

Finally, substitute these values into the equation for \( \frac{ds}{dt} \):
\[
\frac{ds}{dt} = \frac{2000}{3605.6} \cdot 200
\]
\[
\frac{ds}{dt} \approx 0.5546 \cdot 200 = 110.92 \, \text{ft/sec}
\]

### Answer:
At the instant when the airplane is 2000 feet horizontally from the radar station, the distance between the radar station and the airplane is increasing at approximately **110.9 ft/sec**.

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Would you like a more detailed explanation of any step or additional clarification?

Here are five related questions to expand on this concept:

1. How would the result change if the airplane was flying at a higher altitude?
2. What happens to \( \frac{ds}{dt} \) when the airplane is directly above the radar station (i.e., \( x = 0 \))?
3. How would you find the time it takes for the airplane to travel a certain distance horizontally?
4. Can this problem be solved using parametric equations for the airplane’s path?
5. How would the result be different if the airplane’s speed was increasing (i.e., if \( \frac{dx}{dt} \) was a function of time)?

**Tip**: In related rates problems, always make sure to differentiate each variable with respect to time and apply the chain rule properly!

An airplane is flying level at an altitude of 3000 feet and a speed of 200 ft/sec. It is about to pass over a radar station on the ground. How fast is the distance between the station and the airplane changing when the horizontal distance from the station is 2000 feet?

This problem explores how the distance between a radar station and an airplane flying at 3000 feet is changing. Using the Pythagorean Theorem and related rates differentiation, we solve for the rate of change at the moment the airplane is 2000 feet horizontally from the station. Suitable for students in Grades 10-12.

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