A highway patrol plane flies 4 mi above a level, straight road at a steady 190 mph. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 mi, the line-of-sight distance is decreasing at the rate of 165 mph. Find the car's speed along the highway.

Car's speed is_____ miles per hour

To solve this problem, we can use related rates in calculus. Let's define the variables and set up the problem.

### Variables:
- Let \( y = 4 \) miles be the height of the plane above the road (this is constant).
- Let \( x \) be the horizontal distance between the car and the point on the road directly below the plane.
- Let \( s \) be the line-of-sight distance from the plane to the car.
- Let \( v \) be the speed of the car along the highway, which we need to find.

### Given:
- The speed of the plane is 190 mph.
- The line-of-sight distance \( s = 5 \) miles.
- The rate at which the line-of-sight distance is decreasing: \( \frac{ds}{dt} = -165 \) mph (negative because the distance is decreasing).

### Relationships:
The line-of-sight distance forms a right triangle with the horizontal distance \( x \) and the vertical height \( y \). According to the Pythagorean theorem:
\[
s^2 = x^2 + y^2
\]
Differentiating both sides with respect to time \( t \):
\[
2s \frac{ds}{dt} = 2x \frac{dx}{dt}
\]
Simplifying:
\[
s \frac{ds}{dt} = x \frac{dx}{dt}
\]
Substituting \( \frac{dx}{dt} \) with \( v \), the speed of the car:
\[
s \frac{ds}{dt} = x v
\]

### Find \( x \):
When \( s = 5 \) miles and \( y = 4 \) miles, we can find \( x \) using the Pythagorean theorem:
\[
x^2 + 4^2 = 5^2
\]
\[
x^2 + 16 = 25
\]
\[
x^2 = 9 \implies x = 3 \text{ miles}
\]

### Solve for \( v \):
Substituting into the related rates equation:
\[
5(-165) = 3v
\]
\[
-825 = 3v
\]
\[
v = \frac{-825}{3} = -275 \text{ mph}
\]
The negative sign indicates that the car is moving toward the plane. The car's speed along the highway is:

\[
\boxed{275} \text{ mph}
\]

Would you like more details or have any questions?

Here are some related questions:
1. What is the significance of using the Pythagorean theorem in this problem?
2. How would the problem change if the plane's altitude were different?
3. What if the radar indicated the line-of-sight distance was increasing? How would that affect the car's speed?
4. Can this method be applied to other scenarios involving different vehicles or speeds?
5. How would air resistance or wind affect the plane's observations?

**Tip:** When working with related rates, always start by identifying the relationship between variables, and then differentiate with respect to time.

A highway patrol plane flies 4 mi above a level, straight road at a steady 190 mph. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 mi, the line-of-sight distance is decreasing at the rate of 165 mph. Find the car's speed along the highway. Car's speed is_____ miles per hour

Find the speed of a car along a highway using related rates and the Pythagorean theorem. Given a patrol plane's altitude of 4 miles, and a radar showing the line-of-sight distance decreasing at 165 mph, this problem breaks down calculus concepts such as differentiation and related rates. Suitable for high school and early college students.

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