Question

A 1-mile track has parallel sides and equal semicircular ends. Find a formula for the area enclosed by the track, \(A(d)\), in terms of the diameter \(d\) of the semicircles. What is the natural domain for this function?

Short answer

The formula for the area is \( A(d) = \frac{\pi d^2}{4} + \frac{d(1 - \pi d)}{2} \) with domain \([0, \frac{1}{\pi}]\).

Step-by-step solution

Break Down the Problem

The track consists of two parallel straight sections and two semicircular ends. The given parameter is the diameter \(d\) of the semicircles. The length of the straight sections combined with the circumference of the semicircular ends is 1 mile.

Calculate the Length of the Semicircular Ends

The diameter of each semicircle is \(d\), so the radius \(r\) is \(\frac{d}{2}\). The circumference of a full circle is \(2\pi r\). Since we have semicircles, the total length for the semicircular ends is \(\pi r = \frac{\pi d}{2}\). Since there are two semicircular ends, their combined length is \(\pi d\).

Find the Length of Straight Sections

The total length of the track is 1 mile. Since the semicircular ends contribute \(\pi d\) to the total length, the straight sections cover the remaining distance: \(1 - \pi d\). As there are two straight sections, each is \(\frac{1 - \pi d}{2}\) miles long.

Calculate the Area of the Semicircles

The area of a full circle is \(\pi r^2\). Each semicircle has an area \(\frac{\pi r^2}{2}\), so the area of both semicircles combined is \(\pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}\).

Calculate the Area of the Rectangle

The straight sections form the sides of a rectangle that has a length \(\frac{1 - \pi d}{2}\) and width \(d\). Thus, the area of the rectangular portion is \(d \times \frac{1 - \pi d}{2} = \frac{d(1 - \pi d)}{2}\).

Derive the Formula for the Area of the Track

The total area enclosed by the track is the sum of the area of the semicircular ends and the rectangle: \[ A(d) = \frac{\pi d^2}{4} + \frac{d(1 - \pi d)}{2} \].

Determine the Natural Domain of the Function

The diameter \(d\) must be non-negative, and the condition \(\frac{1 - \pi d}{2} \geq 0\) ensures that the straight sections have non-negative length. Solving \(1 - \pi d \geq 0\) gives \(d \leq \frac{1}{\pi}\). Thus, the natural domain is \([0, \frac{1}{\pi}]\).

Key concepts

Track Geometry

When designing or analyzing fenced paths or tracks, understanding their geometry is crucial. In this case, the track is composed of two distinct sections: parallel straight paths and semicircular ends. The parallel sections contribute to the straight, elongated middle of the track, while the semicircular ends create curved sections at both ends.
To solve geometry problems like these, it is essential first to visualize the shape: imagine a looped track with parallel lines flanked by semicircles. This composite shape ensures continuous smooth movement, typically seen in running tracks and cycling velodromes.
For calculations involving the track, note that the linear sections are the only straight lines, whereas the curves contribute significantly to the track's total perimeter. The straight and curved sections together must measure the given specific length, here specified as 1 mile.

Semicircle Calculations

In geometry, semicircles are half-circles which can be quantified with certain key values: diameter, radius, and circumference. The semicircle in this scenario is vital in defining the ends of our track.
- **Diameter**: The exercise gives the diameter, \(d\), of the semicircle. It is the longest straight line that passes through the center of the circle.
- **Radius**: The radius, \(r\), is half the diameter, given as \( \frac{d}{2}\).
- **Circumference**: The length of the arc for the semicircle is half of a circle's, thus \( \frac{\pi d}{2}\) for one semicircle. Thus, two semicircles contribute a total arc length of \( \pi d \) to the track.
Understanding these properties allows us to calculate the contribution of the semicircles to the overall structure, including the perimeter length and area.

Natural Domain of a Function

The natural domain of a function refers to all the possible input values for which the function is defined. In our issue, this concept allows us to determine feasible values for the diameter \(d\) of the semicircles.
To ensure the track is physically possible, the length of the straight sections must be non-negative, which gives us the condition \(1 - \pi d \geq 0\). Solving this inequality, we find that \( d \leq \frac{1}{\pi} \).
Thus, the natural domain of the function \(A(d)\), defined as the area of the track in terms of diameter \(d\), is \([0, \frac{1}{\pi}]\). This ensures that for every value of \(d\) within this interval, our track can exist in real-world conditions.

Areas of Composite Shapes

The track described is a composite shape that combines semicircles and a rectangle, which together create the full track area. Calculating areas for these composite shapes is about adding the areas of individual components.
- **Semicircular Areas**: The area for both semicircles is found using \( \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4} \). This part of the track contributes significantly to the total area within the track.
- **Rectangular Area**: The rectangle's dimensions are determined by the track's straights and the diameter of the semicircles. The length of the rectangle is \( \frac{1 - \pi d}{2} \), and the width equals \(d\). Thus, its area is \( \frac{d(1 - \pi d)}{2} \).
- **Total Area**: Add the areas of the semicircles and rectangle to get the full area of the track: \(A(d) = \frac{\pi d^2}{4} + \frac{d(1 - \pi d)}{2}\). Composite area calculations like these require combining the contributions of differing sections accurately.