An airplane is flying towards a radar station at a constant height of 5 km above the ground. If the distance z between the airplane and the radar station is decreasing at a rate of 380 km per hour when z=20 km what is the horizontal velocity of the plane?
Please enter your answer in decimal format with three significant digits after the decimal point.

This is a related rates problem involving an airplane flying towards a radar station. We are given:

- The constant height \( h = 5 \) km.
- The distance between the airplane and the radar station \( z \) is decreasing at a rate of \( \frac{dz}{dt} = -380 \) km/h when \( z = 20 \) km.
- We need to find the horizontal velocity of the airplane \( \frac{dx}{dt} \) when \( z = 20 \) km, where \( x \) is the horizontal distance between the airplane and the radar station.

### Step 1: Relating the variables
Using the Pythagorean theorem, the relationship between \( x \), \( h \), and \( z \) is:
\[
z^2 = x^2 + h^2
\]
Since \( h = 5 \) km is constant, we substitute that into the equation:
\[
z^2 = x^2 + 25
\]

### Step 2: Differentiate with respect to time
Now, differentiate both sides of the equation with respect to time \( t \):
\[
2z \frac{dz}{dt} = 2x \frac{dx}{dt}
\]
Simplifying:
\[
z \frac{dz}{dt} = x \frac{dx}{dt}
\]

### Step 3: Solve for \( \frac{dx}{dt} \)
We can solve for the horizontal velocity \( \frac{dx}{dt} \) by rearranging the equation:
\[
\frac{dx}{dt} = \frac{z \frac{dz}{dt}}{x}
\]

### Step 4: Find \( x \) when \( z = 20 \) km
When \( z = 20 \) km, we use the Pythagorean theorem to find \( x \):
\[
20^2 = x^2 + 5^2
\]
\[
400 = x^2 + 25
\]
\[
x^2 = 375
\]
\[
x = \sqrt{375} = 19.3649 \, \text{km}
\]

### Step 5: Plug in the known values
Now, substitute \( z = 20 \) km, \( \frac{dz}{dt} = -380 \) km/h, and \( x = 19.365 \) km into the equation for \( \frac{dx}{dt} \):
\[
\frac{dx}{dt} = \frac{(20)(-380)}{19.365}
\]
Let's calculate this:

\[
\frac{dx}{dt} = \frac{-7600}{19.365} \approx -392.411 \, \text{km/h}
\]

Thus, the horizontal velocity of the plane is approximately \( \boxed{-392.411} \) km/h (in decimal format with three significant digits).

Would you like further details on any step?

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Here are five related questions for deeper understanding:
1. How do related rates problems apply to real-life scenarios, like tracking planes?
2. What if the airplane was descending while moving horizontally? How would the height change affect the calculations?
3. How does the Pythagorean theorem help in modeling real-world problems?
4. What would the situation look like if the radar station was also moving?
5. Can you solve similar problems involving other geometric relationships, like circles or spheres?

**Tip:** Always differentiate implicitly with respect to time when solving related rates problems, and don’t forget to account for negative rates if quantities are decreasing.

An airplane is flying towards a radar station at a constant height of 5 km above the ground. If the distance z between the airplane and the radar station is decreasing at a rate of 380 km per hour when z=20 km, what is the horizontal velocity of the plane? Please enter your answer in decimal format with three significant digits after the decimal point.

Learn how to calculate the horizontal velocity of a plane flying towards a radar station using related rates and the Pythagorean theorem. This problem involves differentiation and real-time tracking of rates of change. Suitable for advanced calculus or physics students.

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