Problem 4 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Your 26 inch diameter bicycle wheel has a patch on it. Let \(P\) be the function defined by \(P(x) \quad\) is the distance from patch to the ground where \(x\) is the distance you have traveled on a bicycle trail. a. Draw a graph of \(P\) (approximate is acceptable). b. Find the period and amplitude of \(P\).

Step-by-Step Solution

Answer

The period of \(P(x)\) is approximately 81.68 inches, and the amplitude is 13 inches.

Step 1: Understanding the Problem

In this problem, the function \(P(x)\) represents the distance from the patch on a bicycle wheel to the ground as the bicycle travels along a trail. As the wheel rotates, this distance will vary periodically.

Step 2: Finding the Period

The bicycle wheel has a 26-inch diameter, so its circumference, which is the distance through which it rotates once, is \(C = \pi \times 26 \approx 81.68\) inches. Thus, the period of the function \(P(x)\) is 81.68 inches, as the patch returns to its original position relative to the ground after traveling one full circumference.

Step 3: Finding the Amplitude

The amplitude of the function is half the diameter of the wheel, since this is the maximum distance the patch will be from the ground in either direction. Hence, the amplitude is \(\frac{26}{2} = 13\) inches.

Step 4: Graphing the Function

To graph \(P(x)\), note the function is a sine wave due to the circular motion of the wheel. The graph has a mean value of 13 inches (half the diameter), with a peak (maximum) at 26 inches and a trough (minimum) at 0 inches. It repeats every 81.68 inches. Draw the x-axis as distance, and the y-axis as height from the ground. The completed graph will resemble a sine wave oscillating between 0 and 26 inches with a period of 81.68 inches.

Key Concepts

Amplitude

The amplitude of a periodic function, like the one we're considering in this exercise, is a measure of how much it oscillates up or down from its mean value. In simple terms, it tells us the maximum extent of the wave measured from the center line (mean value).

For the bicycle wheel exercise,

• the diameter is known to be 26 inches.
• The amplitude would therefore be half of this, calculated as \( \frac{26}{2} = 13 \) inches.

This means the distance from the patch to the ground will have a maximum (and minimum) swing of 13 inches from the center line, which makes sense when considering the full rotation of a wheel.

Understanding the amplitude helps us to know how high or low the patch will go from the average position as the wheel turns.

Period

The period of a periodic function is the horizontal distance required for the function to complete one full cycle and return to its starting position. It's crucial in understanding how often the pattern repeats itself.

For the function in the bicycle wheel problem,

• the period is defined by the wheel's circumference.
• With a diameter of 26 inches, the circumference and thus the period is computed using \( C = \pi \times 26 \approx 81.68 \) inches.

This means after traveling 81.68 inches, the patch on the wheel will return to its initial position relative to the ground.

Grasping this concept allows us to predict the consistency of the function as the wheel rotates along the path.

Graphing Functions

Graphing periodic functions, like the function for our bicycle wheel, involves translating theoretical calculations into a visual representation, often helping in comprehending these functions more intuitively.

To graph the function for the bike wheel, consider these steps:

• Axes Setup: Label the x-axis as distance traveled and the y-axis as height from the ground.
• Midline: Mark the midline at 13 inches, equidistant from the maximum and minimum values.
• Range: Oscillations will occur between 0 (lowest point) and 26 inches (highest point).
• Cycle Markers: Position cycle markers at 0, 81.68 inches, and so on, marking where full cycles reset.

Visualizing the sine wave helps in understanding how periodic changes translate as the wheel moves, making it clear how the patch's distance from the ground varies in a regular, repeating pattern.