Problem 89 - Thomas Calculus

Question

The circular waves in the photograph here were made by touching the surface of a ripple tank, first at \(A\) and then at \(B\) . As the waves expanded, their point of intersection appeared to trace a hyperbola. Did it really do that? To find out, we can model the waves with circles centered at \(A\) and \(B .\) (GRAPH NOT COPY) At time \(t,\) the point \(P\) is \(r_{A}(t)\) units from \(A\) and \(r_{B}(t)\) units from \(B .\) since the radii of the circles increase at a constant rate, the rate at which the waves are traveling is $$\frac{d r_{A}}{d t}=\frac{d r_{B}}{d t}$$ Conclude from this equation that \(r_{A}-r_{B}\) has a constant value, so that \(P\) must lie on a hyperbola with foci at \(A\) and \(B\) .

Step-by-Step Solution

Answer

Yes, the point P traces a hyperbola as \( r_A - r_B \) is constant.

Step 1: Understanding the Wave Expansion

As waves expand from points A and B, they form circles with increasing radii. These expansions occur at the same constant rate. We express this rate as \( \frac{d r_A}{d t} = \frac{d r_B}{d t} \), where \( r_A(t) \) and \( r_B(t) \) represent the radii from points A and B, respectively.

Step 2: Setting Up the Constant Rate Equation

Since both rates of expansion are equal, the equations \( r_A(t) = v \cdot t + C_A \) and \( r_B(t) = v \cdot t + C_B \) hold, where \( v \) is the constant speed of the waves and \( C_A, C_B \) are initial radii.

Step 3: Subtracting Radii to Find the Relationship

Subtract the equations to get \( r_A(t) - r_B(t) = (v \cdot t + C_A) - (v \cdot t + C_B) = C_A - C_B \). This difference, \( r_A - r_B \), remains constant over time.

Step 4: Interpretation of r_A - r_B as a Hyperbola Property

In the definition of a hyperbola, for any point \( P \) on the hyperbola, the absolute difference in distances to the foci is constant. Here, A and B serve as the foci with \( |r_A - r_B| \) being constant. Thus, a set of points \( P \) will trace out a hyperbola.

Key Concepts

Hyperbola

A hyperbola is an important curve in the study of conic sections. It is similar to a pair of mirrored parabolas, but its unique property is that it consists of two separate curves called branches. This distinct shape occurs when a plane intersects both halves of a double cone.
To understand how it relates to the exercise, consider that a hyperbola has two fixed points called foci. For any point on the hyperbola, the absolute difference in distances to these two foci is constant. That is exactly what happens in our problem with points A and B.

• Points A and B are the foci of the hyperbola.
• The expanding waves from A and B have their intersection tracing a hyperbola, thanks to the constant difference in radii.

This exercise demonstrates how natural phenomena can illustrate mathematical concepts through visual and spatial representations.

Circular Waves

Circular waves are waves that spread out in circles from a point source. In the exercise, these waves originate from points A and B, forming concentric circles with an increasing radius from each respective starting point. As the waves expand at a constant rate, their radii become crucial in determining the next points of wave intersection.
These intersections form visible patterns, like a hyperbola, when their behavior is modeled and analyzed mathematically.

• The uniform rate of expansion is significant.
• The pattern of intersection relates to how circles overlap as they grow.

Circular waves allow us to connect predictable movement in physics to the precision of geometry.

Rate of Change

The rate of change is a critical aspect of understanding dynamic systems, such as expanding circular waves. In this exercise, the rate of change specifically refers to how fast the radii of the circles are increasing. It is given as \( \frac{d r_A}{d t} = \frac{d r_B}{d t} \), meaning both circles expand at the same constant speed.
This uniform rate highlights that as time progresses, there is a direct, linear relationship in how the radii grow from their respective points. Hence, when subtracted, any changes in individual radii cancel out over time.
This consistency is foundational to proving the constant difference property that allows the hyperbola pattern to emerge.

Distance Formula

The distance formula is vital in analyzing relationships like those in our exercise. It allows us to determine the distance between two points, such as P and the foci of a hyperbola (A and B in this case).
The formula is usually stated as \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) and helps to compute the distances \( r_A \) and \( r_B \).

• This formula transforms spatial relationships into numerical expressions.
• Here, it underpins the proof that the absolute distance difference \( |r_A - r_B| \) is constant as P traces the hyperbola.

Understanding how to measure and calculate these distances fosters deeper insight into the geometric interpretation of movement and change.