A plane flying horizontally at an altitude of 2 miles and a speed of 520 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it has a total distance of 5 miles away from the station. (Round your answer to the nearest whole number.)


To find the rate at which the distance from the plane to the radar station is increasing, we need to set up the problem using related rates. 

Let's denote:
- \( x \) as the horizontal distance from the plane to the radar station.
- \( y \) as the altitude of the plane (which is 2 miles and constant).
- \( z \) as the distance from the plane to the radar station.

Given:
- The plane's speed horizontally, \( \frac{dx}{dt} = 520 \) mi/h.
- The altitude \( y = 2 \) miles.
- The total distance \( z = 5 \) miles.

We want to find \( \frac{dz}{dt} \), the rate at which the distance \( z \) is changing.

Using the Pythagorean theorem, the relationship between \( x \), \( y \), and \( z \) is:
\[ z^2 = x^2 + y^2 \]

Differentiate both sides with respect to \( t \):
\[ 2z \frac{dz}{dt} = 2x \frac{dx}{dt} \]

Simplify by dividing both sides by 2:
\[ z \frac{dz}{dt} = x \frac{dx}{dt} \]

We can solve for \( \frac{dz}{dt} \):
\[ \frac{dz}{dt} = \frac{x}{z} \frac{dx}{dt} \]

We need to find \( x \) when \( z = 5 \) miles:
\[ z^2 = x^2 + y^2 \]
\[ 5^2 = x^2 + 2^2 \]
\[ 25 = x^2 + 4 \]
\[ x^2 = 21 \]
\[ x = \sqrt{21} \]

Now substitute \( x = \sqrt{21} \), \( z = 5 \), and \( \frac{dx}{dt} = 520 \) mi/h into the equation:
\[ \frac{dz}{dt} = \frac{\sqrt{21}}{5} \times 520 \]

Calculate the rate:
\[ \frac{dz}{dt} = \frac{\sqrt{21}}{5} \times 520 \]
\[ \frac{dz}{dt} = \frac{520 \sqrt{21}}{5} \]
\[ \frac{dz}{dt} = 104 \sqrt{21} \]

Using the approximate value of \( \sqrt{21} \approx 4.58 \):
\[ \frac{dz}{dt} \approx 104 \times 4.58 \]
\[ \frac{dz}{dt} \approx 476.32 \]

Rounded to the nearest whole number, the rate at which the distance from the plane to the radar station is increasing is approximately \( 476 \) mi/h.

Would you like details or have any questions?

Here are some related questions for further understanding:
1. How do you differentiate implicit functions?
2. What is the physical significance of related rates in motion problems?
3. How would the problem change if the altitude of the plane varied with time?
4. What if the plane's speed were given in different units, such as feet per second?
5. Can this problem be modeled using vector calculus?
6. How does changing the altitude affect the rate of distance increase?
7. What are other practical applications of related rates in engineering?
8. How can we extend this problem to three-dimensional space? 

**Tip:** For problems involving related rates, always identify the relationship between variables and use differentiation to find the rates of change.

Learn how to find the rate at which the distance from a plane to a radar station is increasing using related rates and the Pythagorean theorem. In this problem, the plane flies horizontally at an altitude of 2 miles and a speed of 520 mi/h. When the plane is 5 miles away from the radar station, discover how to calculate the rate of change in distance. Suitable for advanced high school students and calculus learners.

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