Suppose you're driving down a straight road, a little too fast. There is a police officer positioned 22m off the road who is trying to catch speedsters like yourself. The police officer aims their range finder at your vehicle and the reading indicates that your car is 60 away and that distance is decreasing at a rate of . What is your vehicle's actual speed at that moment? (Your answer should be a positive number.) Give unsimplified final equation

Let's set up the situation and variables to find your car's actual speed:

### Variables
1. Let:
   - \( x \) be the distance between your car and the point on the road closest to the police officer.
   - \( y = 22 \, \text{m} \) be the constant distance between the road and the police officer.
   - \( s \) be the distance between your car and the police officer (the hypotenuse of the triangle formed by \( x \) and \( y \)).
   
2. The problem states:
   - \( s = 60 \, \text{m} \) (distance from the police officer to your car at that instant).
   - \( \frac{ds}{dt} = - \) (the rate at which this distance is decreasing, indicating the officer measures your distance to be getting closer).

### Relationships and Goal
Since we want the actual speed of your car along the road, we're interested in \( \frac{dx}{dt} \), the rate at which \( x \) changes.

Using the Pythagorean theorem, we have:
\[
s^2 = x^2 + y^2
\]

### Step 1: Differentiate with Respect to Time
Differentiating both sides with respect to \( t \):
\[
2s \frac{ds}{dt} = 2x \frac{dx}{dt}
\]

### Step 2: Solve for \( \frac{dx}{dt} \)
Dividing both sides by \( 2x \) gives:
\[
\frac{dx}{dt} = \frac{s \frac{ds}{dt}}{x}
\]

### Step 3: Substitute Values
Since \( y = 22 \) m and \( s = 60 \) m, we can find \( x \) using the Pythagorean theorem:
\[
x = \sqrt{s^2 - y^2} = \sqrt{60^2 - 22^2}
\]

Now, plug in these values to get the unsimplified expression for \( \frac{dx}{dt} \):
\[
\frac{dx}{dt} = \frac{60 \cdot \frac{ds}{dt}}{\sqrt{60^2 - 22^2}}
\]

This expression gives the speed of your vehicle along the road.

Suppose you're driving down a straight road, a little too fast. There is a police officer positioned 22m off the road who is trying to catch speedsters like yourself. The police officer aims their range finder at your vehicle and the reading indicates that your car is 60 away and that distance is decreasing at a rate of . What is your vehicle's actual speed at that moment? (Your answer should be a positive number.) Give unsimplified final equation.

Explore how to calculate a vehicle's speed using related rates and the Pythagorean theorem when a police officer detects a distance of 60 meters with a decreasing rate of approach. This problem involves advanced calculus concepts, including differentiation and related rates, suitable for high school students in Grades 11-12.

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