91Ó°ÊÓ

The Pythagorean theorem is a fundamental principle in geometry. It is used to relate the sides of a right triangle. The formula is expressed as:

• \( a^2 + b^2 = c^2 \)

where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse. In our helicopter scenario, the height (3000 feet) represents one leg, the horizontal distance \( x \) is the second leg, and the line of sight (5000 feet) is the hypotenuse. Using the Pythagorean theorem, we find that:

• \( x^2 = 5000^2 - 3000^2 \)
• \( x = \sqrt{5000^2 - 3000^2} = 4000 \) feet

This relationship helps us understand the geometry of the problem and sets the foundation for further analysis.

Differentiation is a core concept in calculus and involves finding the rate at which a function changes at any given point. When something, such as a function or an angle, changes with time, differentiation helps us calculate how fast this change occurs.
In the solution to our problem, we differentiate the tangent function with respect to time to find the rate of change of the angle \( \theta \). The differentiation of \( \tan(\theta) \) with respect to time \( t \) is expressed as:

• \( \frac{d}{dt}[\tan(\theta)] = \sec^2(\theta) \cdot \frac{d\theta}{dt} \)

This uses the derivative identity, where \( \sec^2(\theta) \) is the derivative of \( \tan(\theta) \). Differentiation allows us to work with continuously changing values, providing insight into how fast or slow something is changing over time.

Rates of change show how a quantity changes over time, often described as speed or velocity in various contexts. In our problem, the focus is on the rate at which the searchlight's angle \( \theta \) changes as the helicopter moves.
To find this rate, we use the relationship established through differentiation. The equation representing this rate of change is:

• \( \sec^2(\theta) \cdot \frac{d\theta}{dt} = \frac{-3000}{x^2} \cdot \frac{dx}{dt} \)

Inserting the known values \( x = 4000 \) feet and \( \frac{dx}{dt} = 100 \) feet per second helps us calculate \( \frac{d\theta}{dt} \), which results in approximately \( \frac{5}{256} \) radians per second.
This value indicates how the searchlight's angle changes over time due to the helicopter's movement.

Trigonometric functions like sine, cosine, and tangent help relate angles to sides in a right triangle. In this scenario, \( \tan(\theta) = \frac{3000}{x} \) relates the vertical height (3000 feet) to the horizontal distance \( x \).
When working with these functions, it is helpful to remember some basic relationships, especially involving the tangent and secant functions:

• \( \tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}} \)
• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)

Trigonometry is key to translating the spatial relationships into mathematical expressions that can be differentiated. It plays an essential role in solving real-world geometry problems, such as determining the speed of a searchlight's rotation.