Question

The sun is just setting in the west as a horse is running around an elliptieal track at the rate of \(m\) miles an hour. The axis of the ellipse lies in the meridian. Find the rate at which the horse's shadow moves on a fence beyond the track and parallel to the axis.

Short answer

Answer: The rate at which the horse's shadow moves on the fence is given by the formula \(\frac{dD}{dt} = \frac{2\pi mb}{T} \cos(\frac{2\pi mt}{T})\) miles per hour, where \(m\) is the horse's speed, \(b\) is the semi-minor axis of the ellipse, and \(T\) is the time it takes for the horse to complete one lap around the track.

Step-by-step solution

Determine the position of the horse at a given time

First, we can find the position of the horse on the elliptical track, \((x(t), y(t))\) with respect to time \(t\). Given that the horse is running around the track at \(m\) miles per hour, we can write this as a parametric equation: \(x(t) = a \cos(\frac{2\pi mt}{T})\) \(y(t) = b \sin(\frac{2\pi mt}{T})\) where \(a\) and \(b\) are the semi-major and semi-minor axes of the ellipse respectively, and \(T\) is the time it takes the horse to complete one lap around the track.

Find the position of the horse's shadow on the fence

Next, we will find the position of the horse's shadow on the fence, \((x_s(t), y_s(t))\). Since the sun is setting in the west, the light rays are almost horizontal. Therefore, we can assume that the angle of elevation is very small. In this case, the x-coordinate of the shadow on the fence should be equal to the x-coordinate of the horse on the track. That is, \(x_s(t) = x(t)\). For the y-coordinate, we notice that as the sun is setting, the shadow on the fence is elongated almost horizontally, and thus \(y_s(t)=0\). Hence, the position of the horse's shadow on the fence is given by \((x_s(t), y_s(t)) = (x(t), 0)\).

Calculate the rate of horse's shadow movement

Now, we will find the rate at which the distance between the horse and its shadow on the fence changes i.e., \(\frac{dD}{dt}\), where \(D\) is the distance between the horse and its shadow. The distance between the horse and its shadow can be given by the Pythagorean theorem: \(D = \sqrt{(x(t) - x_s(t))^2 + (y(t) - y_s(t))^2}\) Since \(x_s(t) = x(t)\) and \(y_s(t) = 0\), we have: \(D = \sqrt{(y(t))^2} = y(t) = b \sin(\frac{2\pi mt}{T})\) Now, we differentiate the distance with respect to time: \(\frac{dD}{dt} = b \cos(\frac{2\pi mt}{T}) \cdot \frac{2\pi m}{T} = \frac{2\pi mb}{T} \cos(\frac{2\pi mt}{T})\) Therefore, the rate at which the horse's shadow moves on the fence is \(\frac{dD}{dt} = \frac{2\pi mb}{T} \cos(\frac{2\pi mt}{T})\) miles per hour.

Key concepts

Elliptical Trajectory

An elliptical trajectory is a path described by an object moving in an ellipse. In our context, the horse runs around an elliptical track, which is a flattened circle with two focal points. Unlike a circular path with a constant radius, an ellipse has a varying distance from its center to any point on its perimeter. This distance is shortest at the minor axis and longest at the major axis.

To describe motion along an elliptical path, we typically use a pair of variables that depend on time, since the horse's position changes constantly as it moves around the track. These variables are encapsulated in parametric equations that separately describe the horizontal and vertical positions of the horse over time.

Parametric Equations

Parametric equations represent a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. Here, the horse's position on the elliptical track \(x(t), y(t)\) is expressed in terms of time \(t\) using the parametric equations.

Parametric equations are powerful tools in calculus because they allow us to describe complex motions, like an elliptical path, in terms of simpler functions. The horse's x-coordinate follows a cosine wave, while the y-coordinate follows a sine wave, with both adjusted for the speed \(m\) and the period \(T\) of one full lap around the track. We can then use these equations to examine different aspects of the motion, such as velocity and acceleration, by applying differentiation to these parametric equations.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

This theorem comes into play when calculating the distance between two points in a plane, such as the horse and its shadow. Even though they share the same x-coordinate, there's a vertical distance between them, represented by the y-coordinate. This creates a right triangle with the horse, its shadow, and the point vertically above the shadow on the fence as vertices. However, since the x-coordinates are the same, the theorem simplifies the distance to the absolute value of the y-coordinate.

Differentiation

Differentiation is a concept in calculus that deals with finding the rate at which a function changes at any given point. It is used to calculate derivatives, which can represent velocity, acceleration, and other rates of change in various contexts.

In our scenario, after determining the horse's distance from its shadow using the Pythagorean theorem, differentiation helps us find the rate at which this distance changes—the speed of the shadow on the fence. By differentiating the distance function with respect to time, we can pinpoint the instantaneous rate of shadow movement at any given moment. This application of differentiation transforms a static distance measurement into a dynamic one, providing us with a deeper understanding of how the horse's movement influences its shadow.