91Ó°ÊÓ

The Pythagorean Theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This is expressed in the formula:\[ c^2 = a^2 + b^2 \]where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.

In our problem involving a highway patrol plane, we use the Pythagorean Theorem to relate the distance between the plane and a car. In this scenario, the hypotenuse \(d\) is the line-of-sight distance from the plane to the car. The vertical component of our right triangle is the altitude of the plane, which is constant at 3 miles. The horizontal component \(x\) is the distance along the road from the plane to the car.

By applying the Pythagorean Theorem, we derive the equation \(d^2 = x^2 + 3^2\). This relationship is crucial for setting up further calculations to find the car's speed.

Differentiation is a key concept in calculus that allows us to find the rate at which a variable changes with respect to another. It is especially useful in dynamic situations where variables are time-dependent, such as in our highway patrol problem.

To calculate how fast the distance \(x\) (the horizontal distance along the road) is changing, we differentiate the original Pythagorean relationship \(d^2 = x^2 + 3^2\) with respect to time \(t\). When we do this implicitly, we get the equation:\[ 2d \frac{dd}{dt} = 2x \frac{dx}{dt} \]
In this expression, \(\frac{dd}{dt}\) represents the rate at which the line-of-sight distance \(d\) is changing. Likewise, \(\frac{dx}{dt}\) represents the rate at which the horizontal distance \(x\) is changing. By substituting known values into this differentiated equation, we reveal the information necessary to solve for the car's speed.

Velocity is a vector quantity that refers to the rate of change of an object's position. It includes both the speed of the object and the direction of its movement. In the context of the highway patrol problem, we are interested in determining the velocity of the car along the straight road.

The given problem provides that the line-of-sight distance is decreasing at 160 miles per hour. This velocity tells us that the car is moving towards the plane. Be careful with the signs: a negative sign in our calculations usually indicates direction (here, towards the plane).

After solving our differentiated equation, we found \( \frac{dx}{dt} = -200 \) miles per hour, indicating that the car's speed is 200 miles per hour towards the plane. The magnitude of this velocity gives us the car's speed along the highway, and the negative sign indicates its direction.

A right triangle is a triangle in which one angle is exactly 90 degrees. This type of triangle is significant in many mathematical applications, including the Pythagorean Theorem.

In the highway patrol plane example, the setup is naturally modeled as a right triangle. The plane, flying parallel to the road at a constant altitude, forms the vertical leg of the triangle. The road, upon which the car travels, is the base or horizontal leg.

The hypotenuse comprises the line-of-sight distance from the plane to the car, essential in tracking changes in distance as the car moves. Understanding this simple geometric structure allows us to apply principes like the Pythagorean Theorem effectively, as it represents real-world scenarios in a manner that can be easily calculated and analyzed.